SAT 2026 Math Test (Get Free Access of Module 2

Question: The combined original price for a mirror and a vase is $60. After a 25% discount to the mirror and a 45% discount to the vase are applied, the combined sale price for the two items is $39. Which system of equations gives the original price $m$, in dollars, of the mirror and the original price $n$, in dollars, of the vase?
A)
$m + v = 60$
$0.55m + 0.75v = 39$
B)
$m + v = 60$
$0.45m + 0.25v = 39$
C)
$m + v = 60$
$0.75m + 0.55v = 39$
D)
$m + v = 60$
$0.25m + 0.45v = 39$

Understand the Question
~ Mirror + Vase original price = $60
~ Mirror discount = 25%
~ Vase discount = 45%
~ Sale price total = $39
Which system represents this?

🧠 Core Concept (Discount logic)
If an item has a 25% discount, you pay:
$$100\% – 25\% = 75\% = 0.75$$
If an item has a 45% discount, you pay:
$$100\% – 45\% = 55\% = 0.55$$

🧮 Step-by-Step Explanation
Original price equation:
$$m + v = 60$$
Sale price equation:
$m$ for mirror (75%$= \frac{75}{100} = 0.75$)

$v$ for vase (55% $= \frac{55}{100} = 0.55$)
$$0.75m + 0.55v = 39$$
✅ Correct Answer: C

Option A ❌
Trap: Student subtracts discounts incorrectly.

Option B ❌
Trap: Student uses discount instead of remaining price.

Option D ❌
Trap: Student uses discount rates directly.

Question: The graph of a system of an absolute value function and a linear function is shown. What is the solution $(x, y)$ to this system of two equations?
A) (-1, 5)
B) (0, 4)
C) (1, 5)
D) (4, 2)

1️⃣ Explain the Question
~ A solution to a system is the intersection point
~ The point must lie on both graphs
~ You must read the exact intersection from the graph

2️⃣ Rules / Concepts Used
System of Equations (Graphical)
~ Each solution = point where graphs cross
~ Only one intersection is shown
~ (x, y): It is absolute.

3️⃣ Step-by-Step Graph Analysis
From the image:
~ The absolute value graph forms a “V”
~ The linear graph crosses it at exactly one point
~ That intersection is clearly at: $(x, y)$
$x = 1$
$y = 5$

✅ Correct Answer: Option C

❌ (-1, 5) → Lies on one graph, not both
❌ (0, 4) → On the line but not on the V
❌ (4, 2) → Vertex of V but no line there

Question:
$4a^2 + 20ab + 25b^2$
Which of the following is a factor of the polynomial above?
A) $a + b$
B) $2a +5b$
C) $4a + 5b$
D) $4a + 25b$

🧠 Core Concept (Recognize structure first)
This matches:$$(A + B)^2 = A^2 + 2AB + B^2$$

🧮 Step-by-Step Solution
Step 1: Arrange into Formula
We have: $4a^2 + 20ab + 25b^2$
$A^2 = 4a^2$ then what is $A = 2a$
$B^2 = 25b^2$ then what is $B = 5b$
$2AB = 20ab$ then its expanded form is $2(A)(B) = 2(2a)(5b)$

Rewrite:
$4a^2 + 20ab + 25b^2\\(2a)^2 + 2(2a)(5b) + (5b)^2\\(2a + 5b)^2$

✅ Correct Answer: B

a + b ❌
Trap: Student ignores coefficients.

4a + 5b ❌
Trap: Student partially factors.

4a + 5b ❌
Trap: Student partially factors.

🧮 DESMOS
1. Type: factor(4a^2 + 20ab + 25b^2)
2. Output: (2a + 5b)^2
$(2a + 5b)^2$

Question: The frequency table summarizes the 57 data values in a data set. What is the maximum data value in the data set?
A) 11
B) 12
C) 14
D) 16

Important Rule
The maximum data value is:
The largest data value with a nonzero frequency

Step-by-Step Solution

We just need to know the Largest Data Value, we do not need to multiply Data Value to its Frequency to find out the largest. Just choose the Largest Data Value.
Data value 14 has frequency 6
Correct Answer$$\boxed{14}$$
✅ Option: 14

Why Other Options Are Incorrect
11 ❌ Not the largest value
12 ❌ Still smaller than 14
16 ❌ This is a frequency, not a data value

Common Student Mistakes
❌ Confusing “how many” with “what value”
❌ Choosing the largest frequency
❌ Ignoring zero frequency

Question: The measure of angle $R$ is $\frac{2 \pi}{3}$ radians. The measure of angle $T$ is $\frac{5 \pi}{12}$ radians greater than the measure of angle $R$. What is the measure of angle $T$, in degrees?
A) 75
B) 120
C) 195
D) 390

Understanding the Question
You are given:
Angle R in radians
Angle T is greater than R by a given radian amount
You must convert the final angle into degrees.

Important Rules
Add radians first
Convert radians → degrees using:$$\text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi}$$

Step-by-Step Solution
Step 1: Given Values$$R = \frac{2\pi}{3}$$T is greater than R, so
$$T = R + \frac{5\pi}{12}$$
Step 2: Add the Radians
Find a common denominator between 3 and 12 LCM:
$\frac{2 \pi}{3} + \frac{5 \pi}{12} \\ \\ \frac{2 \pi \times 4 + 5 \pi \times 1}{12} \\ \\ \frac{8 \pi + 5 \pi}{12} \\ \\ \frac{13 \pi}{12}$

Step 3: Convert to Degrees$$\frac{13\pi}{12} \times \frac{180}{\pi} = \frac{13 \times 180}{12}$$Divide 180 by 12
$$= 13 \times 15 = 195^\circ$$
Correct Answer: Option C$$\boxed{195^\circ}$$

Why Other Options Are Incorrect
75° ❌ Too small, ignores addition
120° ❌ Equals $\frac{2\pi}{3}$, not T
390° ❌ Arithmetic error

Common Student Mistakes
Dropping π incorrectly
Converting before adding
Forgetting common denominators

Use DESMOS as a calculator only.

Question:
$a^2−a−1=0$
What values satisfy the equation?
A) $a = 1$ and $a = 2$
B) $a = -\frac12$​ and $a = \frac32$​
C) $a = \frac{1 + \sqrt{5}}{2}$ and $a = \frac{1 – \sqrt{5}}{2}$
D) $a = \frac{-1 + \sqrt{5}}{2}$​​ and $a = \frac{-1 – \sqrt{5}}{2}$

🧠 Core Concept (Why quadratic formula is needed)
This quadratic:$$a^2 – a – 1 = 0$$
~ does not factor nicely
~ coefficients are small but roots involve irrational numbers
So the quadratic formula is the correct and safest method.

🧮 Step-by-Step Solution
Quadratic formula:$$a = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$
Identify coefficients:
$a = 1$
$b = -1$
$c = -1$

Step 1: Compute the discriminant
$$b^2 – 4ac = (-1)^2 – 4(1)(-1)$$
~ Minus, Minus = Plus, so $-1^2 = -1 \times -1 = 1$
~ Same here = $-4 \times 1 \times -1 = -4 \times -1 = 4$
$$= 1 + 4 = 5$$
Step 2: Substitute into the formula

$a = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

Let’s input the already calculate part:

$a = \frac{-b \pm \sqrt{5}}{2a}$

Now, let’s input other values:

$a = \frac{-(-1) \pm \sqrt{5}}{(2)(1)}$

$a = \frac{1 \pm \sqrt{5}}{2}$

So the solutions are:

$a = \frac{1 + \sqrt{5}}{2},\ \ \ a = \frac{1 – \sqrt{5}}{2}$

✅ Correct Answer: C

Option A ❌
Trap: Student guesses integer roots without checking.

Option B ❌
Trap: Student forces rational roots that don’t satisfy the equation.

Option D ❌
Trap: Student forgets that $-b = +1$, not −1.

Desmos Steps
1. Desmos don’t get $a$ , so it will show nothing: a^2 – a – 1 = 0
2. But it gets x and y
3. Enter: x^2 – x – 1
4. Desmos will graph this equation on the coordinate plane ($a$ on x-axis, y = 0 line).
5. Graph shows:
~ In Desmos, click on the intersection points where the graph crosses the x-axis.
~ (1.61803, 0) = (x, y)
~ (-0.61803, 0) = (x, y)
~ we know, in Desmos, we replace a to x
6. Now solve all options one-by-one to match the intersection points
7. Write options exactly like this – for example Option C
~ 1+ sqrt5 = it gives this 3.2360679775
~ copy it and paste it separate line then type: /2
~ 3.2360679775/2
~ You will get 1.61803…
~ That means it is correct, same step 7 – do it for 1-sqrt5
The intersect points match to Option C, done.

Question: If $\frac{x}{y} = 4$ and $\frac{24x}{ny} = 4$, what is the value of $n$?
A) 4
B) 6
C) 24
D) 96

Important Algebra Rule
If two fractions are equal:$$\frac{x}{y} = \frac{24x}{ny}$$
you can substitute or simplify using known ratios.

Step-by-Step Solution
Step 1: Use the first equation$$\frac{x}{y} = 4 \Rightarrow x = 4y$$
Step 2: Substitute into the second equation
$$\frac{24x}{ny} = 4$$
Replace $x = 4y$:$$\frac{24(4y)}{ny} = 4$$
Step 3: Simplify$$\frac{96y}{ny} = 4$$
Cancel $y$:$$\frac{96}{n} = 4$$
Step 4: Solve for n
$\\ \frac{96}{n} = 4 \\ \\ 96 = 4n \\ \\ n = \frac{96}{4} \\ \\ n = 24$

Correct Answer$$\boxed{24}$$
✅ Option: 24

Why Other Options Are Incorrect
4 ❌ Too small, doesn’t balance equation
6 ❌ Arithmetic mismatch
96 ❌ Forgot to divide by 4

Common Student Mistakes
❌ Cancelling incorrectly
❌ Forgetting substitution
❌ Solving too early

Desmos Verification
1. Type: x/y = 4
2. Type in 2nd line: 24x/ny = 4
3. Desmos will create a graph line.
4. Desmos asks to add a slider or inform the value of n
5. Enter: n value from options one-by-one
6. When you enter n = 24, notice that the question equation graph line and Option C graph line, matches exactly. It overlaps the question graph line.
That is your answer.

Question: Some values of the linear function $f$ are shown in the table above. What is the value of $f(3)$?
A) 6
B) 7
C) 8
D) 9

🧠 Core Concept (Why slope matters)
The question explicitly says linear function.
That tells us one critical fact:
~ The rate of change (slope) is constant everywhere.
So if we find the slope once, we can find any missing value, including $f(3)$.

This table is not about random values.
Each pair $(x, f(x))$ represents a point on the graph of the function:
~ When $x = 0$, $f(x) = -2$ → point (0, −2)
~ When $x = 2$, $f(x) = 4$ → point (2, 4)
~ When $x = 6$, $f(x) = 16$ → point (6, 16)
So we are not inventing anything like “$y_1$​, $y_2$​”.
~ Since it is Math, we can assume $f(x)$ point as $y$.
~ Like $(0, -2) = (x, y)$
We are simply using the definition of a function:
~ A function assigns one output $f(x)$ to each input $x$

🧮 Step-by-Step Solution
Step 1: Find the slope using two points
Using $(0, -2)$ and $(2, 4)$:
~ $x: 0, 2.$
~ $f(x)\ or\ y = -2, 4.$
Slope formula:
$m = \frac{y_2 – x_2}{y_1 – x_1}$ or $m = \frac{change\ in\ f(x)}{change\ in\ x}$

Use any two points from the table (because the function is linear).

$$m = \frac{4 – (-2)}{2 – 0} = \frac{6}{2} = 3$$
So the function increases by 3 units in output for every 1 unit increase in x.

Step 3: What the Slope Means Here (Concept Check)
Slope $m = 3$ means:
~ Increase $x$ by 1 → $f(x)$ increases by 3
~ Increase $x$ by 2 → $f(x)$ increases by 6
This interpretation is essential for the next step.

Step 4: Move From a Known Value to $x = 3$
We already know:
$$f(2) = 4$$
We want:
$$f(3)$$
That is an increase of 1 in $x$.
Since slope = 3:
$$f(3) = 4 + 3 = 7$$
✅ Correct Answer: B) 7

Option A: 6 ❌
Trap: Student assumes slope = 2 instead of 3
(They divide incorrectly or eyeball the table)

Option C: 8 ❌
Trap: Student adds 4 (confuses output difference with input jump)

Option D: 9 ❌
Trap: Student multiplies slope by $x$ instead of applying change

🧮 Desmos Confirmation (Not the Reasoning)
1. Open Desmos
2. Enter a Table (This is critical)
~ In Desmos, click “+” → Table, then enter: [Don’t try to change y into f(x)]
x y
0 -2
2 4
6 16
👉 Desmos will automatically plot the points.
3. Use Linear Regression (Best SAT Trick)
~ In a new expression line, type exactly: y1 ~ mx1 + b
Desmos will output values for:
$m = 3$
$b = -2$
So Desmos confirms:$$f(x) = 3x – 2$$
4. Evaluate $f(3)$ Directly in Desmos: f(3) = 3(3) – 2
Desmon outputs: 7

Question: In the $xy$-plane shown, square $ABCD$ has its diagonals on the $x$– and $y$-axes. What is the area, in square units, of the square?
A) 20
B) 25
C) 50
D) 100

Understanding the Diagram
Square ABCD
Diagonals lie on x- and y-axes
The square is centered at the origin
Coordinates shown:
~ Top point at (0, 5)
~ Right point at (5, 0)
~ Bottom point at (0, –5)
~ Left point at (–5, 0)

Key Geometry Rule
For a square:
Diagonal length $d$
Side length:$$s = \frac{d}{\sqrt{2}}$$
Area:$$A = s^2$$

Step-by-Step Solution
Step 1: Length of Diagonal
From (0,5) to (0,–5):$$d = 10$$
Step 2: Side Length$$s = \frac{10}{\sqrt{2}} = 5\sqrt{2}$$
Step 3: Area$$A = (5\sqrt{2})^2 = 25 \times 2 = 50$$
OR
$s = \frac{10}{\sqrt2} \\ \\ A = (\frac{10}{\sqrt2})^2 \\ \\ A = \frac{100}{2} \\ \\ A = 50$

Correct Answer: Option C$$\boxed{50}$$

Why Other Options Are Incorrect
20 ❌ too small
25 ❌ forgot diagonal relationship
100 ❌ squared diagonal incorrectly

Common Student Traps
Forgetting the √2 factor
Treating diagonal as side

Use DESMOS as a calculator.

Question: If $x^2 = a + b$ and $y^2 = a + c$, which of the following is equal to $(x^2 – y^2)^2$?
A) $a^2 – 2ac + c^2$
B) $b^2 – 2bc + c^2$
C) $4a^2 – 4abc + c^2$
D) $4a^2 – 2abc + b^2c^2$

Important Algebra Rules
1. Difference of squares: $x^2 – y^2 = (a + b) – (a + c)$

2. Square a binomial: $(u – v)^2 = u^2 – 2uv + v^2$

Step-by-Step Solution
Step 1: Substitute$$x^2 – y^2 = (a+b) – (a+c)$$
Simplify: $(a + b) – (a + c)\ \to\ a + b – a – c$
$$x^2 – y^2 = b – c$$
Step 2: Square: We have found $x^2 – y^2$ but the question asked $(x^2 – y^2)^2$
So, we put a square on b – c$$(b – c)^2$$
Expand: Binominal formula$$b^2 – 2bc + c^2$$
Correct Answer$$\boxed{b^2 – 2bc + c^2}$$
✅ Option B

Why Other Options Are Incorrect
Option A: uses $a$, but $a$ cancels out
Option C: introduces extra terms not present
Option D: invalid algebraic structure

Common Student Mistakes
❌ Expanding before simplifying
❌ Forgetting cancellation of $a$
❌ Squaring incorrectly

DESMOS TRICKS
1. Assign values to a, b, and c.
a = 1
b = 2
c = 3
~ You can assign any values.
2. Simplify: You must simplify it yourself to this point – take reference from above solution.
~ Type:
~~ No
(x^2 – y^2)^2 ❌
((a + b) – (a + c))^2 ❌
~~ Yes
((1 + 2) – (1 + 3))^2 ✅
~ Output: 1
3. Type options values one-by-one:
~ Option B: 2^2 – 2*2*3 + 3^2
~ Output: 1
Option B and question equation matches the output.

Question: Line 𝓁 is defined by 3y + 12x = 5. Line n is perpendicular to line 𝓁 in the xy-plane. What is the slope of line n?

[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

1/4 or 0.25 is correct.

🧮 Step-by-Step Correct Solution
The question asks for the slope of line n, but line $n$ is not given directly. Instead, we are told an important relationship: line n is perpendicular to line ℓ.
That means the first necessary step is to find the slope of line ℓ.
Without knowing the slope of line $\ell$, we cannot determine the slope of a line perpendicular to it.
Step 1: Rewrite the given equation in slope form
Given:
$$3y + 12x = 5$$
To identify the slope, we need the equation in slope–intercept form:
$$y = mx + b$$
This form is essential because:
~ The coefficient of $x$ (the number multiplying $x$) is by definition the slope
~ Slope is not directly visible unless the equation is written with y alone on one side
We do not isolate x because:
~ The standard slope formula used in SAT is taken from $y = mx + b$
~ Isolating x would require extra conversion later and hides the slope
So we isolate y deliberately to expose the slope clearly.
Subtract $12x$ from both sides:
$$3y = -12x + 5$$
Now divide every term by 3:
$$y = -4x + \tfrac{5}{3}$$
Step 2: Identify the slope of line $\ell$
From:
$$y = -4x + \tfrac{5}{3}$$
The slope of line $\ell$ is:
$$m_{\ell} = -4$$
This value is critical — everything that follows depends on it.

Step 3: Why perpendicular lines use negative reciprocals
A key SAT rule applies here: If two lines are perpendicular, the product of their slopes is −1
Mathematically:
$$m_{\ell} \cdot m_{n} = -1$$
This rule comes from coordinate geometry and is always used on SAT when:
~ The word perpendicular appears
~ Slopes are involved

Step 4: Find the slope of line $n$
We already know:
$$m_{\ell} = -4$$
Apply the perpendicular-slope rule:
$$(-4)(m_n) = -1$$
Solve for $m_n$​:
$$m_n = \text{negative reciprocal of } -4$$
That means:
$$m_n = \tfrac{1}{4}$$
This process is often described as taking the negative reciprocal:

~ Reciprocal of $-4$ → $-\tfrac{1}{4}$

~ Change the sign → $\tfrac{1}{4}$

✔ This is the slope of line $n$
✔ Divide 1 by 4 and you will get 0.25
✔ So either 0.25 or 1/4 is the correct answer

❌ Trap 1: Answer = −4
Why students choose this:
~ They stop after finding the slope of line $\ell$
~ They forget the question asks for the slope of line n
Mistake:
~ Ignoring the word perpendicular
~ ❌ Trap 1: Answer = −4
Why students choose this:
~ They stop after finding the slope of line $\ell$
~ They forget the question asks for the slope of line n
Mistake:
~ Ignoring the word perpendicular
~ Confusing “opposite slope” with “perpendicular slope”

❌ Trap 2: Answer = −1/4 or -0.25​
Why students choose this:
~ They correctly take the reciprocal
~ They forget to change the sign
Mistake: Incomplete application of the negative-reciprocal rule

🧮 DESMOS CALCULATOR — VERIFICATION METHOD
Step-by-Step in Desmos
1. Open Desmos
2. In Expression Line 1, type: y = -4x + 5/3
(This is line $\ell$)
3. In Expression Line 2, type: y = (1/4)x
(This represents a line with slope $\tfrac{1}{4}$)
4. Zoom in near the intersection
5. Observe:
~ Lines intersect at a right angle
~ Confirms perpendicularity

✔ Confirms slope = $\tfrac{1}{4}$

Question: Percent of Residents Who Earned a Bachelor’s Degree or Higher

A survey was given to residents of all 50 states asking if they had earned a bachelor’s degree or higher. The results from 7 of the states are given in the table above. The median percent of residents who earned a bachelor’s degree or higher for all 50 states was 26.95%. What is the difference between the median percent of residents who earned a bachelor’s degree or higher for these 7 states and the median for all 50 states?
A) 0.05%
B) 0.95%
C) 1.22%
D) 7.45%

Understanding the Question
You are given percentages from 7 states and told that the median for all 50 states is 26.95%.
You must:
~ Find the median of the 7 given values
~ Find the difference between the two medians

Key Rule (Median)
~ The median is the middle value when data is arranged in ascending order
~ For 7 numbers, the median is the 4th number

Step-by-Step Solution
Step 1: List the percentages in ascending order
19.5, 21.9, 25.9, 27.9, 30.1, 35.5, 36.4

Step 2: Identify the median of the 7 states
4th value = 27.9%

Step 3: Compare with the national median
National median = 26.95%

Step 4: Find the difference$$27.9 – 26.95 = 0.95$$
✅ Correct Answer: 0.95%

❌ Why the Other Options Are Wrong
7.45% → comparing max and min, not medians
0.05% → subtracting the wrong values
1.22% → difference between wrong middle values

Use DESMOS only a calculator.

Question: An oceanographer uses the equation $s = \frac{7}{4}p$ to model the speed $s$, in knots, of an ocean wave, where $p$ represents the period of the wave, in seconds. Which of the following represents the period of the wave in terms of the speed of the wave?
A) $p = \frac{4}{7}s$

B) $p = \frac{7}{4}s$

C) $p = \frac{4}{7} + s$

D) $p = \frac{7}{4} + s$

Understand the Question
Problem (understanding the question)
Given:$$s = \frac{7}{4}p$$
Where:
$s$ = speed (knots)
$p$ = period (seconds)
Asked:
👉 Express p in terms of s

Important Rule
To solve for a variable: Undo multiplication using division

Step-by-Step Solution$$s = \frac{7}{4}p$$
Multiply both sides by $\frac{4}{7}$​:$$\frac{4}{7}s = p$$
Rewriting:$$p = \frac{4}{7}s$$
Or
There is another mathematical way:
We need to find p
$s = \frac{7}{4}p \\ s = \frac{7}{4} \times p$
Numerator goes on divide of another side.
Denominator / value in divide goes another side as multiply
So:
$\\ \frac{4}{7}s = p \\ p = \frac{4}{7}s$

Correct Answer$$\boxed{p = \frac{4}{7}s}$$
✅ Option A

Why Other Options Are Incorrect
Option B: keeps the same multiplier — no rearranging
Option C & D: add instead of multiply — violates algebra rules

Common Student Mistakes
❌ Switching variables without isolating
❌ Adding instead of dividing
❌ Forgetting inverse operations

Question: In the figure shown, points $Q, R, S$, and $T$ lie on line segment $PV$, and line segment $RU$ intersects line segment $SX$ at point $W$. The measure of $\angle{SQX}$ is $48^\circ$, the measure of $\angle{SXQ}$ is $86^\circ$, the measure of $\angle{SWU}$ is $85^\circ$, and the measure of $\angle{VTU}$ is $162^\circ$. What is the measure, in degrees, of $\angle{TUR}$?
A) 123
B) 134
C) 145
D) 156

✅ Understand the QUESTION — Multi-Triangle Angle Chase
Given
~ Points $Q, R, S, T$ lie on straight line $PV$
~ Lines intersect at $W$
~ Given angle measures:
$\angle SQX = 48^\circ$
$\angle SXQ = 86^\circ$
$\angle SWU = 85^\circ$
$\angle VTU = 162^\circ$
Find $\angle TUR$

1️⃣ Key Geometry Rules
📌 Triangle Sum Rule$$\text{Angles in triangle} = 180^\circ$$
📌 Linear Pair Rule
Angles on a straight line sum to:$$180^\circ$$
📌 Vertical Angles
Vertical angles are equal

2️⃣ Step-by-Step Logic
🔹 Step 1: What we have and what we need to find:
Find $\angle TUR$, to find out $\angle TUR$, we need $\angle URT$ and $\angle RTU$.
~ To find $\angle RTU$, we have given $\angle VTU$.
~ To find $\angle URT$, we need to find $\angle WRS$.
~ To find $\angle WRS,$ we need to find $\angle SWR$ and $\angle RSW$.
~ ~ To find $\angle SWR$, we have given $\angle SWU$.
~ ~ To find $\angle RSW$, we need to find $\angle QSX$.
~ ~ ~ To find $\angle QSX$, we have given $\angle SQX$ and $\angle SXQ$.
It looks difficult but actually, there are only 3-Key rules here. Once you master this type of Questions. All problems from Line, Angle, and Triangles will be easy for you.

🔹 Step 2: Find $\angle QSX$:
We use Triangle Sum Rule because $\angle QSX, \angle SQX$, and $\angle SXQ$ are angles of one triangle,
So
$Angles\ in\ Triangle = 180^\circ\\ \\ \angle QSX + \angle SQX + \angle SXQ = 180^\circ\\ \\ \angle QSX + 48^\circ + 86^\circ = 180^\circ \\ \\ \angle QSX + 134^\circ = 180^\circ \\ \\ \angle QSX = 180^\circ – 134^\circ \\ \\ \angle QSX = 46^\circ$

🔹 Step 3: Find $\angle RSW$:
Look at the figure, notice that $\angle RSW$ and $\angle QSX$ are on the same vertical angle. Look at the point $S$ on both,
So
$Vertical\ angles\ are\ equal.$
$\angle RSW = \angle QSX \\ \\ \angle RSW = 46^\circ$

🔹 Step 4: Find $\angle SWR$:
Notice on the figure that $\angle SWR$ and $\angle SWU$ are a linear pair. Look at the points $S$ and $W$,
So
$Angle\ on\ a\ straight\ line\ sum\ to\ 180^\circ\\ \\ \angle SWR + \angle SWU = 180^\circ \\ \\ \angle SWR + 85^\circ = 180^\circ \\ \\ \angle SWR = 180^\circ – 85^\circ \\ \\ \angle SWR = 95^\circ$

🔹 Step 5: Find $\angle WRS$:
We use Triangle Sum Rule because $\angle WRS,\ \angle SWR$, and $\angle RSW$ are angles of one triangle,
So
$Angles\ in\ Triangle = 180^\circ\\ \\ \angle WRS + \angle SWR + \angle RSW = 180^\circ\\ \\ \angle WRS + 95^\circ + 46^\circ = 180^\circ \\ \\ \angle WRS + 141^\circ = 180^\circ \\ \\ \angle WRS = 180^\circ – 141^\circ \\ \\ \angle WRS = 39^\circ$

🔹 Step 6: Find $\angle URT$:
Look at the figure, notice that $\angle URT$ and $\angle WRS$ are on the same vertical angle. Look at the point $R$ on both,
So
$Vertical\ angles\ are\ equal.$
$\angle URT = \angle WRS \\ \\ \angle URT = 39^\circ$

🔹 Step 7: Find $\angle RTU$:
Notice on the figure that $\angle RTU$ and $\angle VTU$ are a linear pair. Look at the points $T$ and $U$,
So
$Angle\ on\ a\ straight\ line\ sum\ to\ 180^\circ\\ \\ \angle RTU + \angle VTU = 180^\circ \\ \\ \angle RTU + 162^\circ = 180^\circ \\ \\ \angle RTU = 180^\circ – 162^\circ \\ \\ \angle RTU = 18^\circ$

🔹 Step 8: The Final Part – Find $\angle TUR$:
We use Triangle Sum Rule because $\angle TUR, \angle URT$, and $\angle RTU$ are angles of one triangle,
So
$Angles\ in\ Triangle = 180^\circ\\ \\ \angle TUR + \angle URT + \angle RTU = 180^\circ\\ \\ \angle TUR + 39^\circ + 18^\circ = 180^\circ \\ \\ \angle TUR + 57^\circ = 180^\circ \\ \\ \angle TUR = 180^\circ – 57^\circ \\ \\ \angle QSX = 123^\circ$

✅ Correct Answer is 123: Option A

❌ 134 / 145 → missing vertical angle transfer
❌ 156 → assuming straight-line angle instead of triangle sum

🧮 Desmos?
🚫 Not suitable — requires angle inference, not coordinates.

Question: A certain apprentice has enrolled in 85 hours of training courses. The equation 10x + 15y = 85 represents this situation, where x is the number of on-site training courses and y is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?

[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

🧮 Step-by-Step Solution
This equation represents total training hours, not the number of courses.
Each term follows the structure:
$$(\text{hours per course}) \times (\text{number of courses})$$
From the equation: 10x + 15y = 85
~ x represents on-site course
~ 10x → each on-site course takes 10 hours
~ y represents online course
~ 15y → each online course takes 15 hours
The question does not ask for $x$ or $y$.
It asks for the difference in hours per course. (Key word: How many more)
So we compare the coefficients directly.

Step: Compare course durations
$$15 – 10 = 5$$
(A real trap in question if you do not understand or read it properly. Those who miss that try to solve the equation but that is not what the question asks here. It asks:
How many more hours does each online training course take than each on-site training course?
~ It did not ask “How many more number of courses?”
~ It asks “How many more hours of online training course?”

✅ Correct Answer: 5 hours

❌ Trap 1: Answer = 85
Why students choose this:
They confuse total hours with difference per course.

❌ Trap 2: Answer = 25
Why students choose this:
They add the coefficients instead of comparing them:
$$10 + 15$$

❌ Trap 3: Answer = 1.5
Why students choose this:
They divide coefficients instead of subtracting:$$\frac{15}{10}$$

🧮 DESMOS CONFIRMATION
1. Open Desmos
2. Type: 10x + 15y = 85
3. Observe coefficients directly — they represent hours per course
✔ Confirms difference is 5 hours

Question: $(x – 2)^2 = -6$
How many distinct real solutions does the given equation have?
A) Exactly one
B) Exactly two
C) Infinitely many
D) Zero

🔑 Key Algebra Rule
The square of any real number is never negative
$$(\text{real number})^2 \ge 0$$

🧠 Step-by-Step Solution
Left side:$$(x – 2)^2 \ge 0$$
Right side:$$-6 < 0$$
A nonnegative number cannot equal a negative number.
So:$$(x – 2)^2 = -6$$
has no real solution.
✅ Correct Answer – Option D : Zero​

❌ Why Other Options Are Incorrect
Infinitely many ❌ false
Exactly one ❌ impossible
Exactly two ❌ no real roots

Question:
$(ax + 3)(5x^2 – bx + 4) = 20x^3 – 9x^2 – 2x + 12$
The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?
A) 18
B) 20
C) 24
D) 40

1️⃣ What the Question Is Testing
This is polynomial identity matching.
When two polynomials are equal for all x:
👉 Coefficients of like powers of x must be equal
This is not solving for x, this is comparing coefficients.

2️⃣ Step-by-Step Expansion
Expand the left side carefully.$$(ax + 3)(5x^2 – bx + 4)$$
Distribute term by term:
Multiply by $ax$$$ax(5x^2) = 5ax^3$$$$ax(-bx) = -abx^2$$$$ax(4) = 4ax$$
Multiply by $3$$$3(5x^2) = 15x^2$$$$3(-bx) = -3bx$$$$3(4) = 12$$
3️⃣ Combine Like Terms
$5ax^3 – abx^2 + 4ax + 15x^2 – 3bx + 12\\ \\ 5ax^3 + (-abx^2 + 15x^2) + (4ax -3bx) + 12$

4️⃣ Match with the Right Side
Right side:$$20x^3 – 9x^2 – 2x + 12$$
Now match power by power:
$5ax^3 + (-abx^2 + 15x^2) + (4ax -3bx) + 12\ =\ 20x^3 – 9x^2 – 2x + 12$
Now solve it using matching values / figures.

$x^3$ terms: $5ax^3 = 20x^3$
Remove $x^3$ from both sides: $5a = 20$
$\\a = \frac{20}{5}\\ \\a = 4$

$x^2$ terms: $(-abx^2 + 15x^2) = -9x^2$
Remove $x^2$ from both sides: $-ab + 15 = -9$
$-ab = -15 – 9\\ \\ -ab = -24$
Remove negative from both sides: $ab = 24$

We already have the value. We know ab, so we do not need to calculate anymore, but if it is something else, this is how you proceed.

✅ Correct Answer: Option C
$ab = 24$

• 18 / 20 → students forget to include +15 from distribution
• 40 → students match only $5a = 20$ and multiply randomly
• SAT loves coefficient-matching traps

Use Desmos as a calculator only for this one.

Question: A manufacturing plant makes 10-inch, 9-inch, and 7-inch frying pans. During a certain day, the number of 10-inch frying pans that the manufacturing plant makes is 4 times the number n of 9-inch frying pans it makes, and the number of 7-inch frying pans it makes is 10. During this day, the manufacturing plant makes 100 frying pans total. Which equation represents this situation?
A) $10(4n) + 9n + 7(10) = 100$
B) $10n + 9n + 7n = 100$
C) $4n + 10 = 100$
D) $5n + 10 = 100$

Understand the Question
Problem (understanding the situation)
A manufacturing plant makes three sizes of frying pans:
10-inch pans
9-inch pans
7-inch pans

Given:
Number of 9-inch pans = n
Number of 10-inch pans = 4n
Number of 7-inch pans = 10
Total pans made = 100
The question asks:
👉 Which equation correctly represents this situation?

Important Concept / Rule
This is a word-to-equation translation problem.
Key idea: When counting objects, we add the number of objects, not their sizes.
⚠️ The inches (10-inch, 9-inch, 7-inch) are labels, NOT values to multiply.

Step-by-Step Construction of the Equation
10-inch pans → 4n pans
9-inch pans → n pans
7-inch pans → 10 pans
Total pans:$$4n + n + 10 = 100$$
Simplify:$$5n + 10 = 100$$
Correct Answer:$$\boxed{5n + 10 = 100}$$✅ Option D

Why Other Options Are Incorrect
Option A:$$10(4n) + 9n + 7(10)$$
❌ Incorrectly multiplies pan size with quantity
Pan size does NOT affect the count

Option B:$$10n + 9n + 7n$$
❌ Assumes all quantities are proportional to size — false

Option C:$$4n + 10 = 100$$
❌ Ignores the n 9-inch pans

Common Student Mistakes
❌ Multiplying by pan sizes
❌ Forgetting one category
❌ Treating inches as quantities

Question: Near the end of a US cable news show, the host invited viewers to respond to a poll on the show’s website that asked, “Do you support the new federal policy discussed during the show?” At the end of the show, the host reported that 28% responded “Yes,” and 70% responded “No.” Which of the following best explains why the results are unlikely to represent the sentiments of the population of the United States?
A) The percentages do not add up to 100%, so any possible conclusions from the poll are invalid.
B) Those who responded to the poll were not a random sample of the population of the United States.
C) There were not 50% “Yes” responses and 50% “No” responses.
D) The show did not allow viewers enough time to respond to the poll.

🔑 Key Statistical Concept
This is a sampling bias question.
A valid survey requires:
~ Random selection
~ Equal chance for all members of the population to respond

🧠 Step-by-Step Explanation
The poll was taken from viewers of one cable news show
Only people who:
~ Watch that show
~ Visit the website
~ Choose to respond
were included
This means the sample:
~ Is self-selected
~ Is not random
~ Likely reflects opinions of a specific audience, not the US population

✅ Correct Answer
B) Those who responded to the poll were not a random sample of the population of the United States.​

❌ Why Other Options Are Incorrect
Option A
❌ Percentages not totaling 100% could be rounding — not the main issue.

Option C
❌ Surveys do not need 50–50 results to be valid.

Option D
❌ Time allowed does not fix sampling bias.

Question:
$ax + by = 72 \\ 6x + 2by = 56$
In the given system of equations, $a$ and $b$ are constants. The graphs of these equations in the xy-plane intersect at the point (4, y). What is the value of $a$?
A) 14
B) 6
C) 4
D) 3

Choice A is correct.
🧮 Step-by-Step Solution
~ xy-plane (x, y) = (4, y)
~ Because the point of intersection is $(4, y)$,
~ both equations must be true when x = 4.
That is the key instruction hidden in the question.

$ax + by = 72 \\ 6x + 2by = 56$
Step 1: Substitute $x = 4$ into the first equation
$$a(4) + by = 72$$
$$4a + by = 72$$
Rearrange:
$$by = 72 – 4a$$
Step 2: Substitute $x = 4$ into the second equation
$$6(4) + 2by = 56$$
$$24 + 2by = 56$$
Subtract 24:
$$2by = 32$$
Divide by 2:
$$by = 16$$
Step 3: Equate the two expressions for $by$
From above:
$$72 – 4a = 16$$
Solve:
$$4a = 56$$
$$a = 14$$
✅ Correct Answer: A) 14

Option B: 6 ❌
Trap: Student divides 56 by 4 directly, ignoring system structure.

Option C: 4 ❌
Trap: Student confuses $x = 4$ with $a = 4$.

Option D: 3 ❌
Trap: Student divides 12 by 4 after incorrect simplification.

🧮 DESMOS CONFIRMATION
1. Open Desmos
2. Enter:
ax + by = 72
6x + 2by = 56
3. Substitute $x = 4$ and test values of $a$
4. Only a = 14 gives a common intersection point
✔ Confirmed

Question: The product of two positive integers is 462. If the first integer is 5 greater than twice the second integer, what is the smaller of the two integers?

[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

🔹 Understand the QUESTION
Given
~ The product of two positive integers is 462
~ The first integer is 5 greater than twice the second integer
~ Find the smaller integer

🧠 STEP 1: DEFINE VARIABLES (WHY THIS SETUP)
There are two integers.
Let the second integer: The smaller integer = $x$
Then the first integer is: Twice the second integer = $2x$ with 5 greater
~ First integer = $2x + 5$
This setup is chosen because:
~ SAT word problems always describe one quantity in terms of another
~ “5 greater than twice” means multiply first, then add

🧮 STEP 2: USE THE PRODUCT CONDITION
The product is 462, so: 2nd Integer(1st Integer) = Total
$$x(2x + 5) = 462$$
Expand:$$2x^2 + 5x = 462$$
Move everything to one side:$$2x^2 + 5x – 462 = 0$$
This step is necessary because:
~ Quadratic equations must be written in standard form to factor or solve
Method 1:
$\\ 2x^2 + 5x – 462 = 0$
For quadratic, we need to prepare these first: $ax^2 + bx + c = 0$
we will turn b into this $ax^2 + (b_1 + b_2)x + c = 0$ then to $(ax^2 + b_1) + (b_2 + c) = 0$
After that just simplify it:
$\\ ax = 2x \\ \\ax^2 = 2x^2\\ \\bx = 5x$
$b = 5 = (b_1 + b_2)$ , so the best values suit here are (-28 + 33) = 5
$\\ b_1 = -28\\ \\ b_2 = 33\\ \\c = -462$

Here is a twist the value c made out of 2 integer, so $b_1 \times b_2$ must be product of 2c:
$2c = 2(-462) = -924$, so $(b_1 \times b_2) = (-28 \times 33) = -924$
Now, let’s put it all together:

Factor the equation:$$(2x + 33)(x – 14) = 0$$
$(2x + 33) = 0$ then $2x + 33 = 0$ then $2x = -33$ then $x = – \frac{33}{2}$
$(x – 14) = 0$ then $x – 14 = 0$ then $x = 14$

So:$$x = -\frac{33}{2} \quad \text{or} \quad x = 14$$
Method 2:
$\\ 2x^2 + 5x – 462 = 0$

The formula: $x=\frac{-b\pm \sqrt{b^2 – 4ac}}{2a}$

Put values: $x=\frac{-5\pm \sqrt{5^2-4 \times 2 \times -462}}{2 \times 2}$

$x=\frac{-5\pm \sqrt{25 + 3696}}{4}$

$x=\frac{-5\pm \sqrt{3721}}{4}$

$x=\frac{-5\pm 61}{4}$

$\pm$ this sign means, we can do either plus or minus, so:

$x=\frac{-5 + 61}{4}$ or $x=\frac{-5 – 61}{4}$

$x=\frac{56}{4}$ or $x=\frac{-66}{4}$

$x=14$ or $x= – \frac{33}{2}$

Both Method 1 and 2 are correct, choose according to your choice.

🚫 STEP 3: APPLY CONTEXT RESTRICTIONS
The question states positive integers, so:
~ Fractions are not allowed
~ Negative values are not allowed
~ and we need a smaller integer
Thus:$$x = 14$$
Verifying:
1. Input x = 14 into the equation: $x(2x + 5) = 462$
$\\ 14(2 \times 14 + 5) = 462\\ \\14(28 + 5) = 462\\ \\14(33) = 462\\ \\462 = 462$

⚠️ COMMON TRAPS (REAL MISTAKES)
❌ Forgetting integers must be whole numbers
❌ Forgetting “twice” and writing $x + 5$
❌ Solving the quadratic but choosing the negative root
❌ Giving the larger integer instead of the smaller

🧮 DESMOS METHOD (REAL & VALID)
Step 1: Open Desmos
Type: x(2x + 5) = 462
Step 2: Convert to y-form
y = x(2x + 5) – 462
Step 3: Find x-intercepts
Desmos shows: Go to graph and only focus on x-axis
x ≈ −16.5
x = 14
Only 14 satisfies the conditions.

Question: The function $k$ is defined by $k(x) = \frac{|x|}{a} – 14$, where $a < 0$. What is the product of $k(15a)$ and $k(7a)$?

[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

Given
$$k(x) = \frac{|x|}{a} – 14 \quad \text{where } a < 0$$
Find:$$k(15a) \times k(7a)$$
✅ Step-by-Step Mathematical Solution
🔹 Step 1: Understand the key conditions

~ If $k(x) = \frac{|x|}{a} – 14$, then:
$k(x) = k(15a)$ and $k(7a)$, it means
$x = 15a$ and $7a$

Recall the absolute value rule:
This condition controls how absolute value behaves.
$$|u| = \begin{cases} u, & u \ge 0 \\ – u, & u < 0 \end{cases}$$
u is just to demonstrate, you can replace it with a or anything else.
~ If $a < 0$, then: $|a| = -a$
$15a$ and $7a$
So the $a$ in both inputs is a negative numbers.

🔹 Step 2: Evaluate $k(15a)$
Start with the definition:$$k(15a) = \frac{|15a|}{a} – 14$$
Absolute Value Handling (Critical Concept)
Since:
15 is positive
$a < 0$
Then:$$|15a| = 15|a|$$
And because $a < 0$, we know:$$|a| = -a$$
So:$$|15a| = 15(-a) = -15a$$
Substitute back:$$k(15a) = \frac{-15a}{a} – 14$$
Cancel $a$ (allowed since $a \neq 0$):
$$k(15a) = -15 – 14 = -29$$

🔹 Step 3: Evaluate $k(7a)$
Same logic, new input:$$k(7a) = \frac{|7a|}{a} – 14$$
Because $a < 0$:$$|7a| = -7a$$
So:$$k(7a) = \frac{-7a}{a} – 14$$
Cancel $a$:$$k(7a) = -7 – 14 = -21$$
🔹 Step 4: Find the Product
$$k(15a) \times k(7a) = (-29)(-21)$$
$$= 609$$
✅ Final Answer: 609

❌ Trap 1: Treating 15a and 7a as “just a”
Wrong — they are inputs, not constants.

❌ Trap 2: Forgetting a < 0
If you assume a > 0, you will get completely wrong signs.

❌ Trap 3: Dropping absolute value incorrectly
Absolute value changes everything when negatives are involved.

🧮 REAL Desmos Method (Correct Usage Only)
This is how a student would actually verify the answer in Desmos.
Step 1: Define the function (using a slider)
1. Type: k(x) = abs(x)/a – 14
2. Create a slider for a and set:
a = -1 (or any negative value)
~ It means, type any value the answer will come same here but it must be negative.
SAT allows this because the expression depends on sign, not magnitude.

Step 2: Evaluate the function values
1. In Desmos, type:
k(15a)
k(7a)
2. Desmos will display:
k(15a) = -29
k(7a) = -21

Step 3: Multiply directly
1. Type: k(15a) * k(7a)
or
just directly type: -29 * -21
2. Desmos output: 609
✅ Confirms the algebraic solution perfectly.