SAT Math Module 1st (How to Get 1500+ Hack, Free Test 2025

Question: The function $f$ is defined by $f(x) = 25x + 30$. What is the value of $f(x)$ when $x = 2$?
A) 110
B) 80
C) 57
D) 50

Choice B is correct. Substitute x = 2 is already given, all we have to do is to put the value of x into the equation.
f(x) = 25(x) + 30
f(x) = 25(2) + 30
f(x) = 50 + 30
f(x) = 80.

Choice A is incorrect. This is the value of (25 +30)(2), not 25(2) + 30.

Choice C is incorrect. This is the value of 25 + 2 + 30, not 25(2) + 30.

Choice D is incorrect. This is the value of 25(2), not 25(2) + 30.

🧮 DESMOS METHOD (VERY PRECISE)
Method 1: Table (Fastest)
1. Open Desmos
2. In Expression Line 1, type: f(x) = 25x + 30
3. Click the Table icon
4. Enter: x = 2
5. Desmos shows: f(x) = 80
✅ Confirmed

Method 2: Graph Reading
1. Keep: y = 25x + 30
2. Click on the point at $x = 2$
3. Desmos displays: (2, 80)

Question: There are 55 students in Spanish club. A sample of the Spanish club students was selected at random and asked whether they intend to enroll in a new study program. Of those surveyed, 20% responded that they intend to enroll in the study program. Based on this survey, which of the following is the best estimate of the total number of Spanish club students who intend to enroll in the study program?
A) 11
B) 20
C) 44
D) 55

✅ Understand the QUESTION
Estimation from a Sample (Statistics)
Given:
~ Total students = 55
~ Survey result = 20% intend to enroll
Asked:
Best estimate of total students intending to enroll.

🧠 Step-by-Step Solution
Step 1: Convert percent to decimal$$20\% = 0.20$$

Step 2: Multiply by total students$$0.20 \times 55 = 11$$
✅ Correct Answer: Option A

20 ❌
Mistakes percent for number of students.

44 ❌
Uses 80% instead of 20%.

55 ❌
Assumes everyone enrolls.

🧮 Desmos Check
1. Type: 20/100
2. You will get: 0.2
3. Multiply: 0.2*55
4. Output: 11
Or
You can directly type this: 20/100 * 55

Question:
$6x − 9y > 12$
Which of the following inequalities is equivalent to the inequality above?
A) $x – y > 2$
B) $2x – 3y > 4$
C) $3x – 2y > 4$
D) $3y – 2x > 2$

🧠 Core Concept (Why “equivalent” matters)
Two inequalities are equivalent if:
~ they describe the same solution region
~ one can be obtained from the other by legal algebraic operations
(adding, subtracting, multiplying/dividing by a positive number)

🧮 Step-by-Step Solution
We start with:
$$6x – 9y > 12$$
Step 1: Look for a common factor
All coefficients (6, −9, 12) are divisible by 3.
Dividing by 3 simplifies the inequality without changing its direction
(because 3 is positive).
$$\frac{6x – 9y}{3} > \frac{12}{3}$$
$$2x – 3y > 4$$
✅ Correct Answer: B) $2x − 3y > 4$

Option A: $x − y > 2$ ❌
Trap: Student divides incorrectly by 6 instead of 3.

Option C: $3x−2y>4$ ❌
Trap: Student swaps coefficients incorrectly.

Option D: $3y−2x>2$ ❌
Trap: Student rearranges terms and forgets to flip signs.

🧮 Desmos Confirmation (Correct Way)
Enter both: together option-by-option
6x – 9y > 12
2x – 3y > 4
The shaded regions overlap exactly, confirming equivalence.

Question: $f(x) = 4x + b$
For the linear function $f$, $b$ is a constant and $f(7) = 28$. What is the value of $b$?
A) 7
B) 0
C) 1
D) 4

Option B: It is given $f(x) = f(7)$. So substitute $x = 7$:
f(x) = 4x + b
28 = 4(7) + b
28 = 28 + b
28 – 28 = b
0 = b or b = 0

Choice A is incorrect and may result from conceptual or calculation errors. Assuming b = x.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

🧮 DESMOS METHOD
1. Expression Line: 4(7) + b = 28
2. Solve → b = 0

Question:
$3x^2 − 15x +18 = 0$
How many distinct real solutions are there to the given equation?
A) Exactly one
B) Exactly two
C) Infinitely many
D) Zero

🧠 Core Concept (Quadratic roots logic)
A quadratic equation:
$$ax^2 + bx + c = 0$$
has:
~ no real solutions if the discriminant $< 0$
~ one real solution if the discriminant $= 0$
~ two real solutions if the discriminant $> 0$

🧮 Step-by-Step Solution
Method 1:
Step 1: Factor the equation (why factoring first)
$3x^2 − 15x +18 = 0$
All terms share a factor of 3.
$$3(x^2 – 5x + 6) = 0$$
Step 2: Factor the quadratic
$(x^2 − 5x +6)$
Rules & Steps: (Ignore 3)
~ $(x – a) (x – b)$
~ $x = x\ (not\ x^2)$
~ $a + b = 5$
~ $a \times b = 6$
Choose a number for $a\ and\ b$
~~ when multiply, it gets 6 = $2 \times 3$
~~ when do addition, it gets 5 = $2 + 3$
$$x^2 – 5x + 6 = (x – 2)(x – 3)$$
Step 3: Solve
$$x = 2 \quad \text{or} \quad x = 3$$
These are two different real numbers.
✅ Correct Answer: Exactly two

Method 2:
🔑 The Discriminant Rule (This Is the Formula Used)
The discriminant is:$$\Delta = b^2 – 4ac$$
And the rules are:
~ If $\Delta > 0$ → exactly two distinct real solutions
~ If $\Delta = 0$ → exactly one real solution
~ If $\Delta < 0$ → zero real solutions
This rule is absolute, no exceptions.
We are given: $ax^2 + bx + c = 0$$$3x^2 – 15x + 18 = 0$$
Identify coefficients carefully (this step matters):
$a = 3$
$b = -15$
$c = 18$
Step 1: Compute the discriminant: $Δ = b^2 – 4ac$
$$\Delta = (-15)^2 – 4(3)(18)$$
$$\Delta = 225 – 216$$
$$\Delta = 9$$
Step 2: Interpret the value
$$\Delta = 9 > 0$$
Positive discriminant means:
~ the graph crosses the x-axis twice
~ the equation has two different real x-values
✅ Correct Answer: Exactly two

Exactly one ❌
Trap: Student assumes repeated root without checking factorization.

Infinitely many ❌
Trap: Would only occur if equation reduced to $0 = 0$.

Zero ❌
Trap: Student assumes quadratic has no real roots by default.

🧮 Desmos Confirmation
Enter: y = 3x^2 – 15x + 18
Graph crosses x-axis at x = 2 and x = 3.

Question: The perimeter of triangle ABC is 17 inches, the length of side AB is 4 inches, and the length of side AC is 7 inches. What is the length, in inches, of side BC?
A) 4
B) 6
C) 7
D) 11

✅ Understand the QUESTION — Triangle Perimeter
Question Explained
You are given:
~ Total perimeter of triangle $ABC = 17$ inches
~ Side $AB = 4$ inches
~ Side $AC = 7$ inches
You are asked to find the remaining side BC.

📐 Important Rule / Formula
Triangle Perimeter Formula$$\text{Perimeter} = AB + BC + AC$$
This is a definition, not a trick.

🧠 Step-by-Step Solution
Substitute known values:$$17 = 4 + BC + 7$$
Combine known sides:$$17 = 11 + BC$$
Subtract 11 from both sides:$$BC = 6$$
✅ Correct Answer: Option B

❌ Why Other Options Are Incorrect
~ 4 → assumes two sides are equal (not stated)
~ 7 → repeats an already given side
~ 11 → adds known sides instead of subtracting from perimeter
⚠️ Common Student Mistake: forgetting that perimeter means sum of all three sides, not just two.

🧮 Desmos Trick (Simple Check)
1. Type: 17 – (4 + 7)
2. Desmos outputs 6

Question:
$y = 76$
$y = x^2 – 5$
The graphs of the given equations in the xy-plane intersect at the point $(x, y)$. What is a possible value of $x$?

A) $\frac{-75}{5}$

B) -9
C) 5
D) 76

Mathematical solution (algebra first)
At an intersection point, both y-values are equal, so: $y = y$
$$x^2 – 5 = 76$$
Add 5 to both sides or just move $-5$ to right-hand side:
$$x^2 = 81$$
Take square roots: $x = \sqrt{81}$
$$x = \pm 9$$
After root the sign can be either positive or negative.
From the options:
~ The −9 is present
~ The +9 is NOT listed in option

Correct answer: −9

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of coordinate y, rather than x, of the intersection point (x, y).

Desmos – real intersection method
1. Open desmos.com/calculator
2. Enter:
y = 76
y = x^2 - 5
3. Desmos automatically shows intersection points
4. Click the intersection dots
5. Desmos displays:
(−9, 76)
(9, 76)
6. Match with options → −9
✔ This is the standard SAT Desmos intersection trick

Question:
$d = 16 – \frac {x}{30}$
The equation shown gives the estimated amount of diesel $d$, in gallons, that remains in the gas tank of a truck after being driven $x$ miles, where 0 ≤ $x$ ≤ 480. What is the estimated amount of diesel, in gallons, that remains in the gas tank of the truck when $x$ = 300?
A) 0
B) 6
C) 14
D) 16

Choice B is correct.
🧮 Step-by-Step Solution
We are asked to evaluate the function at a specific value, so we directly substitute $x = 300$.
Step 1: Substitute
$$d = 16 – \frac{300}{30}$$
Step 2: Simplify division first
$$\frac{300}{30} = 10$$
Step 3: Subtract
$$d = 16 – 10 = 6$$
✅ Correct Answer: B) 6

Option A: 0 ❌
Trap: Student subtracts incorrectly and assumes empty tank.

Option C: 14 ❌
Trap: Student subtracts $300 ÷ 60$ instead of $300 ÷ 30$.

Option D: 16 ❌
Trap: Student forgets to subtract and uses starting amount.

🧮 DESMOS CONFIRMATION
1. Open Desmos
2. Type: d = 16 – x/30
3. Use Table icon
4. Enter: x = 300
5. Output shows: d = 6
✔ Confirmed

Question: What is the y-intercept of the graph shown?
A. (-1, -9)
B. (0, -5)
C. (0, -4)
D. (0, 0)

Mathematical solution
~ The y-intercept is the point where the graph crosses the y-axis.
~ On the y-axis, x = 0 by definition.
~ (x, y)
~ Looking carefully at the graph, when x = 0, the graph crosses the y-axis at y = −5.
~ Therefore, the y-intercept is (0, −5).
Correct answer: (0, −5)

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at y = -4, not y = -5.

Choice D is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at y = 0, not y = -5.

Question: A rectangle has a length of 3 units and a width of 39 units. Which expression gives the area, in square units, of this rectangle?
A) $2(3 + 39)$
B) $2(3 \cdot 39)$
C) $3 + 39$
D) $3 \cdot 39$

✅ Understand the QUESTION — Area of a Rectangle
Question Explained
You are given:
~ Length = 3 units
~ Width = 39 units
You are asked for an expression that gives the area.

📐 Important Rule / Formula
Area of a Rectangle$$\text{Area} = \text{length} \times \text{width}$$
This is a must-know SAT formula.

🧠 Step-by-Step Solution
Substitute values:$$\text{Area} = 3 \times 39$$
Since the question asks for an expression, not a numerical value, we keep it as:$$\boxed{3 \cdot 39}$$
✅ Correct Answer: Option D

❌ Why Other Options Are Incorrect
~ 2(3+39) → perimeter formula, not area
~ 2(3 ⋅ 39) → doubles the area incorrectly
~ 3+39 → adds dimensions (no geometric meaning)
⚠️ SAT Trap: confusing area with perimeter.

Question: Which expression is equivalent to $(9x^3 + 5x + 7) + (6x^3 + 5x^2 – 5)$?
A) $15x^3 + 5x^2 + 5x + 2$
B) $15x^6 + 5x^2 + 5x + 2$
C) $15x^3 + 10x^2 + 2$
D) $15x^6 + 5x^2 – 5x – 35$

🧠 Core Concept (Why “like terms” rule applies)
Only terms with the same variable AND same exponent can be combined.

🧮 Step-by-Step Algebra
Group like terms: $(9x^3 + 5x + 7) + (6x^3 + 5x^2 – 5)$
$9x^3 + 6x^3 = 15x^3$
$5x^2$ (only appears once)
$5x$ (only appears once)
$7 – 5 = 2$
So the result is:$$15x^3 + 5x^2 + 5x + 2$$
✅ Correct Answer: A

Option B ❌
Trap: Student adds exponents instead of coefficients.

Option B ❌
Trap: Student adds exponents instead of coefficients.

Option D ❌
Trap: Student applies subtraction instead of addition.

🧮 ✅ PROPER DESMOS METHOD
1. Enter both expressions separately
A = 9x^3 + 5x + 7
B = 6x^3 + 5x^2 – 5
2. Add them
A + B
Desmos simplifies automatically to: 15x^3 + 5x^2 + 5x + 2
$15x^3 + 5x^2 + 5x + 2$

Question: Line 𝑝 is defined by 2𝑦 + 18𝑥 = 9. Line 𝑟 is perpendicular to line 𝑝 in the 𝑥𝑦-plane. What is the slope of line 𝑟?
A) -9
B) $-\frac{1}{9}$
C) $\frac{1}{9}$
D) 9

Choice C is correct.
🧮 Complete Step-by-Step Solution
The question asks for the slope of line r, but line $r$ is not given directly.
Instead, we are told it is perpendicular to line $p$.
That means the only possible way forward is:
1. First find the slope of line p
2. Then use the perpendicular-slope rule to find the slope of line $r$

Step 1: Why we isolate y
The equation of line $p$ is:
$$2y + 18x = 9$$
To identify slope, we must write the equation in slope–intercept form:
$$y = mx + b$$
This form is essential because:
~ The slope is explicitly visible as the coefficient of $x$
~ Perpendicular slope rules apply directly to $m$
That is why we isolate y, not $x$.

Step 2: Isolate $y$
Start with:
$$2y + 18x = 9$$
Subtract $18x$ from both sides:
$$2y = -18x + 9$$
Now divide every term by 2:
$$y = -9x + \tfrac{9}{2}$$
Step 3: Identify the slope of line $p$
From:
$$y = -9x + \tfrac{9}{2}$$
The slope of line $p$ is:
$$m_p = -9$$
This value controls everything that follows.

Step 4: Apply the perpendicular-line rule
For perpendicular lines:
$$m_p \cdot m_r = -1$$
Substitute $m_p = -9$:
$$(-9)(m_r) = -1$$
Solve for $m_r$​:
$$m_r = \tfrac{1}{9}$$
This is also described as taking the negative reciprocal:
~ Reciprocal of $-9$ → $-\tfrac{1}{9}$
~ Change the sign → $\tfrac{1}{9}$
✅ Correct Answer: C) $\tfrac{1}{9}$

Option A: −9 ❌
Trap: Student stops after finding the slope of line $p$p and forgets the word perpendicular.

Option B: −1/9 ❌
Trap: Student takes the reciprocal but forgets to change the sign.

Option D: 9 ❌
Trap: Student removes the negative sign but does not take the reciprocal.

🧮 DESMOS CONFIRMATION
1. Open Desmos
2. Enter: 2y + 18x = 9
3. Enter: y = (1/9)x
4. Observe the lines intersect at a right angle
✔ Confirms slope of line $r$

Question: What is the value of $cos \frac{565 \pi}{6}$?
A) $\frac{1}{2}$
B) $1$
C) $\frac{\sqrt{3}}{2}$
D) $\sqrt{3}$

✅ Understand the QUESTION — Trigonometry & Periodicity
Question Explained
You are asked to find:$$\cos\left(\frac{565\pi}{6}\right)$$
📐 Important Rules
1. What the Question Is Really Asking
You are asked to find the cosine of a very large angle measured in radians.
The trick is not to calculate it directly, but to use a fundamental trigonometric rule about cosine.

2. Cosine Period$$\cos(\theta) = \cos(\theta + 2\pi k)$$
What this means (plain English):
~ The cosine graph repeats itself every 2π.
~ After a full rotation around the circle, cosine comes back to the same value.
~ So angles that differ by $2\pi, 4\pi, 6\pi,\dots$ all have the same cosine.
Why 2π?
~ A full circle = 360°
~ In radians, 360° = 2π
~ So cosine “resets” every $2\pi$
This is called the period of cosine, and:$$\boxed{\text{Period of } \cos x = 2\pi}$$
Why Are We Allowed to Use $2\pi$ Here?
Your angle is:$$\frac{565\pi}{6}$$​
This is way larger than 2π, meaning the angle has gone around the circle many times.
👉 Instead of tracking every rotation, we remove full circles using multiples of $2\pi$.

3. Reduce the Angle
Convert to a standard angle between 0 and 2π.

4. Unit Circle Exact Values
From the unit circle:

These are memorized facts, not calculated.

🧠 Step-by-Step Solution
Since:$$2\pi = \frac{12\pi}{6}$$
Divide:$$565 \div 12 = 47 \text{ remainder } 1$$
So:$$\frac{565\pi}{6} = 47(2\pi) + \frac{\pi}{6}$$
If you think, why did we make this:
$\\ \frac{565\pi}{6} = 47(2\pi) + \frac{\pi}{6} \\$
then let’s revert it back to understand.
$\\ 47(2\pi) + \frac{\pi}{6} \\ \\ \frac{47(2\pi)}{1} + \frac{\pi}{6} \\ \\ \frac{47 \times 6 (2\pi) + \pi}{6} \\ \\ \frac{282(2\pi) + \pi}{6} \\ \\ \frac{564\pi + \pi}{6} \\ \\ \frac{565\pi}{6}$
That is why, both are the same. If you think, why did we use 47, because above notice, we calculated 47 remainder 1, means $47(2\pi)$ and remainder $1\pi$ .

Thus: $cos(\theta + 2\pi k) = cos(\theta)$
$\\ cos(47(2\pi) + \frac{\pi}{6}) = cos(\frac{\pi}{6})$
That is WHY
$$\cos\left(\frac{565\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right)$$
And:$$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$$

Why is $\cos\left(\dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}$?
Unit Circle Explanation (Visual Logic)
~ On the unit circle, cosine is the x-coordinate
~ At $\dfrac{\pi}{6}$​ (30°), the point is: (x, y)

$x = \frac{\sqrt{3}}{1} \\ \\ y = \frac{1}{2}$$$\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$$
So:$$\cos\left(\dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}$$
This comes from a 30-60-90 triangle:

✅ Correct Answer: Option C​​​

❌ Why Other Options Are Incorrect
1/2​ → It is the result of cosine of $\frac{\pi}{3}$ or sine of $\frac{\pi}{6}$
1 → cosine of 0
$\sqrt3$​ → impossible (cosine ≤ 1)
⚠️ SAT Trap: not reducing the angle before evaluating cosine.

🧮 Desmos Trick
1. Type: cos(565π/6)
2. Type from 2nd line all the options one-by-one: sqrt3 / 2
3. Observe: The Option C will show the same as the question.
4. Output: 0.8660…
It means Option C is correct.

Question:
$y = 5x + 4$
$y = 5x^2 + 4$
Which ordered pair is a solution to the given (x, y) system of equations?
A) (0, 0)
B) (0, 4)
C) (8, 44)
D) (8, 84)

MATHEMATICAL SOLUTION (CORE LOGIC)
A point is a solution to a system only if it satisfies both equations at the same time.
Since both equations equal y, the logical move is to set them equal to each other.
This step is chosen because both expressions represent the same y-value.
$$5x + 4 = 5x^2 + 4$$
Now simplify:
Subtract 4 from both sides:$$5x = 5x^2$$
Divide both sides by 5:$$x = x^2$$
Rearrange:$$x^2 – x = 0$$
Factor:$$x(x – 1) = 0$$
$x = 0\ ,\ (x-1) = 0\\ x = 0\ ,\ x-1 = 0\\ x = 0\ ,\ x = 1$

So:$$x = 0 \quad \text{or} \quad x = 1$$
Now find the corresponding y-values.
For $x = 0$
Let’s pick: $y = 5x + 4$
$$y = 5(0) + 4 = 4$$
When x is 0, y will be 4.
Point: (0, 4)
For $x = 1$:
$$y = 5(1) + 4 = 9$$
When x is 1, y will be 9.
Point: (1, 9) → not in options
But (0, 4) is in Option B.

❌ (0, 0)
Trap: Assuming origin works by default
~ Plug in x = 0 → y should be 4, not 0
~ Fails both equations

❌ (8, 44)
Trap: Checking only the linear equation
~ 5(8) + 4 = 44 ✔
~ But $5(8^2) + 4 = 324$ ❌
~ Fails quadratic equation

❌ (8, 84)
Trap: Incorrect squaring or mental math
~ Neither equation gives y = 84 for x = 8
Fails both

REAL DESMOS METHOD
1. What to type in Desmos:
y = 5x + 4
y = 5x^2 + 4
2. What Desmos shows:
~ A line and a parabola
~ Their intersection points are the solutions
Click the intersection:
~ Desmos shows (0, 4) and (1, 9)
Only (0, 4) appears in the options.
✔ This directly confirms the algebra.

Question: f(x) = 39
For the given linear function 𝑓, which table gives three values of 𝑥 and their corresponding values of 𝑓(𝑥)?
A)

B)

C)

D)

Choice B is correct.
🧮 Step-by-Step Solution
The function:
$$f(x) = 39$$
means:
~ The output is always 39
~ It does not depend on x
~ The graph is a horizontal line
So for any value of x:
$$f(x) = 39$$
That includes:
$x = 0$
$x = 1$
$x = 2$

Step: Evaluate explicitly$$f(0) = 39$$
$$f(1) = 39$$
$$f(2) = 39$$
✅ Correct Answer: B)

Option A ❌
Trap: Student assumes constant function means zero output.

Option C ❌
Trap: Student incorrectly multiplies:
$$f(x) = 39x$$
instead of
$$f(x) = 39$$

Option D ❌
Trap: Student assumes the function changes sign with x.

🧮 DESMOS CONFIRMATION
1. Open Desmos
2. Type: y = 39
3. Click Table
4. Enter: x = 0, 1, 2
5. All outputs show: 39
✔ Confirmed

Question: The population density of Iceland, in people per square kilometer of land area, increased from 2.5 in 1990 to 3.3 in 2014. During this time period, the land area of Iceland was 100,250 square kilometers. By how many people did Iceland’s population increase from 1990 to 2014?
A) 330,825
B) 132,330
C) 125,312
D) 80,200

Question Understanding
The problem gives population density (people per square kilometer) at two different years and a fixed land area.
You are asked to find how many people the population increased by.
So this is: Population = Density × Land Area

Important Formula / Rule
$$\text{Population} = (\text{people per km}^2) \times (\text{area in km}^2)$$
Population increase:$$(\text{New density} – \text{Old density}) \times \text{Area}$$

Step-by-Step Solution
Step 1: Find the change in population density
New density – Old density
$$3.3 – 2.5 = 0.8 \text{ people per km}^2$$
Step 2: Multiply by land area
Difference in density multiplied by Area:
$$0.8 \times 100,250 = 80,200$$
Correct Answer: Option D$$\boxed{80,200}$$

Why Other Options Are Wrong
330,825 → uses full density instead of the increase
132,330 → arithmetic mistake
125,312 → random incorrect multiplication

Common Student Mistake
❌ Calculating total population instead of population increase

Desmos Trick
In Desmos: (3.3 – 2.5) * 100250

Question: The ratio 140 to $m$ is equivalent to the ratio 4 to 28. What is the value of $m$?
A) 112
B) 114
C) 980
D) 3,290

Question Understanding
You are given two equivalent ratios and must find the missing value $m$.
$$\frac{140}{m} = \frac{4}{28}$$
Important Rule
Equivalent ratios ⇒ cross multiplication
$$\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$$

Step-by-Step Solution
Step 1: Simplify the known ratio
$$\frac{4}{28} = \frac{1}{7}$$
Step 2: Set ratios equal$$\frac{140}{m} = \frac{1}{7}$$
Step 3: Cross multiply$$140 \times 7 = m$$$$m = 980$$
Correct Answer: Option C$$\boxed{980}$$

Why Other Options Are Wrong
112 → incorrect scaling
144 → unrelated value
3,920 → multiplied instead of dividing

Common Student Mistake
❌ Not simplifying ratios before solving

Desmos Trick
1. Type: 140 / x = 4 / 28
~ Desmos will not understand m. It only gets x and y.
2. Check the graph: Zoom out if you are too zoomed in.
~ You will see a straight line on x = 980.
~ That is your answer.

Question: The function $h$ is defined by $h(x) = 4x + 28$. The graph of $y = h(x)$ in the $xy$-plane has an $x$-intercept at ($a$, 0) and a$y$-intercept at (0, $b$), where $a$ and $b$ are constants. What is the value of $a + b$?
A) 21
B) 28
C) 32
D) 35

Choice A is the correct answer.

🧮 Step-by-Step Solution
We are asked for a + b, so we must find:
~ the x-intercept value $a$
~ x-intercept ($a$, 0) = (x₁, x₂)
~ the y-intercept value $b$
~ y-intercept (0, $b$) = (y₁, y₂)
~ the graph y = h(x)
Both intercepts come directly from the equation $y = 4x + 28$, (y = mx + b) but each intercept is found differently, so we must handle them one at a time.

Step 1: Find the y-intercept $b$
Let’s pick up (x₂ and y₂)
~ x₂ = 0
~ y₂ = $b$
By definition, the y-intercept occurs where:
$$x = 0$$
Substitute $x = 0$ into the function: y = 4x + 28
$$y = 4(0) + 28$$
$$y = 28$$
So:$$b = 28$$
Step 2: Find the x-intercept $a$
Let’s pick up (x₁ and y₁)
~ x₁ = $a$
~ y₁ = 0
By definition, the x-intercept occurs where:
$$y = 0$$
Set the function equal to 0: y = 4x + 28
$$0 = 4x + 28$$
Subtract 28 from both sides:
$$-28 = 4x$$
Divide both sides by 4 or move 4 to left-hand side into divide form:
$$x = -7$$
So:$$a = -7$$
Step 3: Compute $a + b$
$$a + b = -7 + 28 = 21$$
✅ Correct Answer: 21

Option B: 28 ❌
Trap: Student finds only the y-intercept and forgets the x-intercept.

Option C: 32 ❌
Trap: Student adds $4 + 28$ instead of intercept values.

Option D: 35 ❌
Trap: Student uses $7 + 28$ instead of $-7 + 28$, ignoring the negative sign.

🧮 DESMOS CONFIRMATION
1. Open Desmos
2. Type: y = 4x + 28
3. Click the x-intercept → shows $(-7, 0)$
4. Click the y-intercept → shows $(0, 28)$
5. ✔ Confirms $a = -7$, $b = 28$

Question: Residents of a town were surveyed to determine whether they are satisfied with the concession stand at the local park. A random sample of 200 residents was selected. All 200 responded, and 87% said they are satisfied. Based on this information, which of the following statements must be true?
I. Of all the town residents, 87% would say they are satisfied with the concession stand at the local park.
II. If another random sample of 200 residents were surveyed, 87% would say they are satisfied.
A) Neither
B) I only
C) II only
D) I and II

Question Understanding
This question tests what can and cannot be concluded from a random sample.
~ Sample size = 200
~ Sample result = 87% satisfied

Important Statistical Rule
A sample percentage:
~ Estimates the population
~ Does not guarantee exact population results
~ Will vary from sample to sample

Evaluate Each Statement
Statement I: Of all town residents, 87% are satisfied.
❌ Not guaranteed
~ Sample ≠ entire population
~ Because they only selected 200 people, there could be more.

Statement II: Another sample of 200 would give exactly 87%.
❌ Not guaranteed
~ Sampling variability exists
~ If the question did not say anything about Another sample, we cannot assume ourselves.

Correct Answer: Option A$$\boxed{\text{Neither}}$$

Why Other Choices Are Wrong
I only → assumes certainty from a sample
II only → ignores random variation
I and II → both false

Common Student Mistakes
❌ Treating sample results as exact population values
❌ Forgetting randomness causes variation

Question: A system of equations consists of a quadratic equation and a linear equation. The equations in this system are graphed in the xy-plane above. How many solutions does this system have?
A) 0
B) 1
C) 2
D) 3

🧠 Core Concept (What “solution” means here)
A solution to a system =
👉 a point where both graphs intersect
So:
~ Each intersection point = one solution
~ We are counting intersection points, not roots

🔍 Step-by-Step Graph Analysis (Why this works)
From the graph:
~ The curve is a parabola opening upward
~ The straight line cuts through the parabola
Now observe carefully:
~ The line intersects the parabola twice
~~ once on the left side
~~ once on the right side
These are two distinct intersection points.
✅ Correct Answer: 2

0 ❌
Trap: Student thinks line never touches parabola.

1 ❌
Trap: Student only notices one intersection and misses the second.

3 ❌
Trap: Student assumes vertex also counts as intersection (it doesn’t).

Question: Jennifer bought a box of Crunchy Grain cereal. The nutrition facts on the box state that a serving size of the cereal is $\frac{3}{4}$ cup and provides 210 calories, 50 of which are calories from fat. In addition, each serving of the cereal provides 180 milligrams of potassium, which is 5% of the daily allowance for adults. If $p$ percent of an adult’s daily allowance of potassium is provided by $x$ servings of Crunchy Grain cereal per day, which of the following expresses $p$ in terms of $x$?
A) $p = 0.5x$
B) $p = 5x$
C) $p = (0.05)^x$
D) $p = (1.05)^x$

Note: The question is made to confuse students, too many unnecessary information but we must focus on only useful ones like below.

✅ Understand the QUESTION
Percent, Unit Rate & Linear Relationship
Given information (from the nutrition label):
~ 1 serving provides 180 mg potassium
~ This equals 5% of an adult’s daily allowance
~ x = number of servings per day
~ p = percent of daily allowance from $x$ servings
Question:
Which expression gives $p$ in terms of $x$?

🧠 Step-by-Step Reasoning
Step 1: Understand what 5% means here
The key sentence is:
“each serving of the cereal provides 180 milligrams of potassium, which is 5% of the daily allowance”
This tells us directly:
~ 1 serving → 5%
~ This is a linear relationship, not exponential because we were not asked “How many Times” here.

Step 2: Scale up to $x$ servings
If:
~ 1 serving gives 5%
~ Then $x$ servings give:$$p = 5 \times x$$
✅ Correct Equation: Option B$$\boxed{p = 5x}$$

p = 0.5x ❌
Confuses 5% with 0.5%.
SAT trap: decimal misplacement.

p = $(0.05)^x$ ❌
Represents exponential decay, which is incorrect.
Potassium accumulates additively.

p = $(1.05)^x$ ❌
Represents compound growth.
SAT trap: mistaking repeated addition for exponential growth.

Question: What system of linear equations is represented by the lines shown?
A) 8x + 4y = 32
-10x – 4y = -64

B) 4x + 10y = 32
-8x – 10y = -64

C) 4x – 10y = 32
-8x + 10y = -64

D) 8x – 4y = 32
-10x + 4y = -64

Option B: (Graph shows two decreasing straight lines intersecting at the x-axis around $x = 8$)

🔍 Step-by-Step Graph Analysis
From the graph, we clearly observe:
• Both lines are sloping downward (negative slope)
• The two lines intersect exactly on the x-axis
• The point of intersection is approximately (x, y) = (8, 0)
• One line is steeper than the other
• The y-intercepts look close to 3–4 for one line and 6–7 for the other
So the correct system must satisfy all of these features.

🧮 Check Option B Carefully (Why It Works)
Method 1:
Equation 1: 4x + 10y = 32
Convert to slope-intercept form: 10y = −4x + 32
y = -4x + 32 divided by 10
y = −0.4x + 3.2

• Negative slope ✔
• y-intercept ≈ 3.2 ✔
• Matches the lower line in the graph ✔

Equation 2: −8x − 10y = −64
Multiply by −1: 8x + 10y = 64 (This step is optional)
Solve for y: 10y = −8x + 64
y = −0.8x + 6.4 divided by 10
y = −0.8x + 6.4

• Negative slope ✔
• Steeper than the first line ✔
• y-intercept ≈ 6.4 ✔

Intersection Check (CRITICAL)
Set equations equal:
y of Equation 1 = y of Equation 2
You can choose any equation to find out x. The x will be the same.
−0.4x + 3.2 = −0.8x + 6.4

The x value matches to the graph, now let’s check y as well.

Method 2: The Quickest Method (Use this when only one value is exactly the same in both like here 10y)
4x + 10y = 32
-8x – 10y = -64

The x value matches to the graph, now let’s check y as well.
Substitute $x = 8$: Now let’s try to fill the value of x and confirm the value of y. Again, the answer will be the same in both equation. For now, we are picking Equation 1.
y = -0.4x + 3.2
y = −0.4(8) + 3.2
y = -3.2 + 3.2
y = 0

✔ Intersection point = (8, 0)
✔ y-intercepts are 3.2 and 6.4 → graph also shows this, something between (3-4) and (6-7)
✔ Exactly matches the graph
✅ Option B is correct

Option A ❌
First equation: 8x + 4y = 32
⇒ 4y = -8x + 32
⇒ y = -8x + 32 divided by 4
⇒ y = −2x + 8
[If you calculate like above, the value of x will be: x = 4. It doesn’t match with the graph. It is incorrect, but if you still want to try for knowledge then continue.]

Second equation: −10x − 4y = −64
⇒ -4y = 10x – 64
⇒ y = 10x – 64 divided by -4
(If in doubt, how to divide then look above the “Without Optional Step.”)
⇒ y = −2.5x + 16
[Here the x will be: x = 6.4. Based on graph, the x should be 8.]

[All you need to do is find x value then calculate y, just like above. The same steps like Option B.]
• y-intercepts are 8 and 16 → graph does NOT show this (
• x is 4 and 6.4 → graph does NOT show this
• y is 0, matches with graph but without x. It is incorrect.
• Slopes are too steep
• Intersection is NOT at (8, 0)
❌ Rejected

Option C ❌ It is a Trap.
Equation 1: 4x − 10y = 32
⇒ y = 0.4x − 3.2
[Here x-intercept is 8. Calculate x like Option B. Let’s find out y.]
y = 0.4(8) – 3.2
⇒ y = 3.2 – 3.2
⇒ y = 0
(Since the Equation 1 matches with the graph, let’s confirm by checking Equation 2, because both should give the same value.)

Equation 2: -8x + 10y = -64
⇒ y = 0.8x – 6.4
[Here x-intercept is 8. Calculate x like Option B. Let’s find out y.]
⇒ y = 0.8(8) – 6.4
⇒ y = 6.4 – 6.4
⇒ y = 0
⚠️ Same as Option B

The REAL Deciding Factor (This Is the Key Point)
The question is NOT: “Which system intersects at (8, 0)?” (If it was then we focus on final values.)
The question is: “Which system matches the LINES SHOWN?”
That means we must match:
– intersection
– direction of slope
– orientation of the lines

From the EQUATION (Algebra Rule): y = mx + b
Option B – Both equations are -0.4x + 3.2 = -0.8x + 6.4
m = -0.4 (Equation 1), -0.8 (Equation 2)
📌 Both slopes are NEGATIVE (The definition of Negative slope: as x increases, y decreases.)
📌 Lines go downward left to right
📌 Exactly matches the graph

Option C – Both equations are 0.4x – 3.2 = 0.8x – 6.4
m = 0.4 (Equation 1), 0.8 (Equation 2)
📌 Both slopes are POSITIVE
📌 Lines go upward left to right
📌 This is the OPPOSITE of the graph
❌ Rejected

Option D ❌
Equation 1: 8x − 4y = 32
⇒ y = 2x − 8
[Here x-intercept is 4. Calculate x like Option B. Since it is clearly incorrect but still let’s find out y.]
y = 2(4) – 8
⇒ y = 8 – 8
⇒ y = 0
(The Equation 1 doesn’t match with the graph.)

• Positive slope
• Does not match graph direction
❌ Rejected

Memory rule for students: On graph if
Left high → right low = negative slope
Left low → right high = positive slope

The sign of m decides direction: y = mx + b
Slope (m) is Positive (+) = Line direction (Upward left → right)
Slope (m) is Negative (−) = Line direction (Downward left → right)

🧮 DESMOS CALCULATOR — EXACT SAT METHOD

Step-by-Step in Desmos
1. Open Desmos Calculator
2. In Expression Line 1, type: 4x + 10y = 32
→ Desmos automatically graphs the line
3. In Expression Line 2, type: -8x – 10y = -64
4. Click on the intersection point shown on the graph
→ Desmos displays: (8, 0)
5. Zoom in using mouse scroll / zoom buttons
6. Confirm:
– Both slopes go downward
– One line is steeper
– Intersection lies exactly on x-axis
✅ Desmos confirms Option B