SAT Math Practice

Q1 Linear Equations Slope ★★☆

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KEY: slope = rise ÷ run = (y₂−y₁)÷(x₂−x₁) Find slope from table → write equation → find x-intercept (set y=0)

x	y
18	130
23	160
26	178

For line h, the table shows three values of x and their corresponding values of y. Line k is the result of translating line h down 5 units in the xy-plane. What is the x-intercept of line k?

Step-by-Step Explanation

First, find the slope of line h using any two table points:

slope = (160 − 130) / (23 − 18) = 30 / 5 = 6

Write the equation of line h using point (18, 130):

y − 130 = 6(x − 18) → y = 6x − 108 + 130 → y = 6x + 22

Line k is h shifted down 5 units, so subtract 5 from the y-intercept:

k: y = 6x + 22 − 5 → y = 6x + 17

Find the x-intercept by setting y = 0:

0 = 6x + 17 → x = −17/6

✅ Answer D: $\left(-\dfrac{17}{6},\ 0\right)$

Q2 Linear Equations Translation ★☆☆

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KEY: vertical shift = add/subtract from y-intercept only Slope NEVER changes when you shift up/down

Line $p$ has equation $y = 4x - 3$. Line $q$ is the result of translating line $p$ up 7 units. What is the y-intercept of line $q$?

Step-by-Step Explanation

When a line is translated up k units, add k to the y-intercept. The slope stays the same.

q: y = 4x + (−3 + 7) = 4x + 4

✅ Answer B: y-intercept is $(0,\ 4)$

Q3 Slope-Intercept x-intercept ★☆☆

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KEY: x-intercept → SET y = 0 and solve for x y-intercept → SET x = 0 and solve for y

Line $m$ has equation $y = -3x + 9$. What is the x-intercept of line $m$?

Step-by-Step Explanation

Set y = 0:

0 = −3x + 9 → 3x = 9 → x = 3

✅ Answer C: $(3,\ 0)$

Q4 Parallel Lines Slope ★★☆

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KEY: parallel = SAME slope · perpendicular = NEGATIVE RECIPROCAL slope e.g. if slope = 2/3, perpendicular slope = −3/2

Line $n$ passes through $(2,\ 5)$ and is parallel to the line $y = 3x - 7$. What is the equation of line $n$?

Step-by-Step Explanation

Parallel lines share the same slope. Slope = 3. Use point (2, 5):

y − 5 = 3(x − 2) → y = 3x − 6 + 5 → y = 3x − 1

✅ Answer B: $y = 3x - 1$

Q5 Systems Infinite Solutions ★★★

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KEY: infinitely many solutions = equations are IDENTICAL (same line) no solution = PARALLEL lines (same slope, different intercept)

\[2x + 3y = 7\] \[10x + 15y = 35\] For each real number $r$, which of the following points lies on the graph of each equation?

Step-by-Step Explanation

The second equation is exactly 5 times the first, so they represent the same line — infinitely many solutions.

From equation 1: $2x + 3y = 7$, solve for y in terms of x:

3y = 7 − 2x → y = (7 − 2x)/3 = −2x/3 + 7/3

If x = r, then y = −2r/3 + 7/3.

✅ Answer C: $\left(r,\ -\dfrac{2r}{3}+\dfrac{7}{3}\right)$

Q6 Systems Substitution ★★☆

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KEY: substitution method = isolate one variable → plug into other equation elimination method = add/subtract equations to cancel a variable

\[y = 2x - 1\] \[3x + y = 9\] What is the value of $x$ in the solution to the system above?

Step-by-Step Explanation

Substitute y = 2x − 1 into the second equation:

3x + (2x − 1) = 9 → 5x − 1 = 9 → 5x = 10 → x = 2

✅ Answer B: $x = 2$

Q7 Systems No Solution ★★★

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KEY: no solution → same slope, DIFFERENT y-intercept (parallel lines) Check: if coefficients are proportional but constants are not → no solution

\[4x - 6y = 10\] \[2x - 3y = k\] For what value of $k$ does the system have no solution?

Step-by-Step Explanation

Divide equation 1 by 2: $2x − 3y = 5$. For the system to have no solution, equation 2 must be parallel — same left side but a different constant.

If k = 5 → same line (infinitely many solutions)

If k ≠ 5 → parallel lines (no solution)

Any value except 5 gives no solution. Among the choices, only k = 7 satisfies k ≠ 5.

✅ Answer B: $k = 7$

Q8 Parabola Tangent Line ★★★

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KEY: y = c intersects parabola at exactly ONE point → discriminant = 0 Set equal → rearrange → b²−4ac = 0 → this is the VERTEX value

In the xy-plane, the graph of $y = -x^2 + 9x - 100$ intersects the line $y = c$ at exactly one point. What is the value of $c$?

Step-by-Step Explanation

The horizontal line y = c touches the parabola at exactly one point when c equals the vertex y-value.

For $y = ax^2 + bx + c_0$, vertex x = −b/(2a):

x = −9 / (2 × −1) = 9/2

y = −(9/2)² + 9(9/2) − 100 = −81/4 + 162/4 − 400/4 = −319/4

✅ Answer C: $c = -\dfrac{319}{4}$

Q9 Quadratic Factoring ★★☆

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KEY: roots = x-values where y = 0 → factor or use quadratic formula If x² + bx + c = (x+p)(x+q), then p+q = b and p×q = c

What are the solutions to $x^2 - 5x + 6 = 0$?

Step-by-Step Explanation

Find two numbers that multiply to 6 and add to −5: those are −2 and −3.

(x − 2)(x − 3) = 0 → x = 2 or x = 3

✅ Answer B: $x = 2$ and $x = 3$

Q10 Vertex Form Completing the Square ★★★

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KEY: vertex form y = a(x−h)² + k → vertex is (h, k) Complete the square: x²+bx = (x + b/2)² − (b/2)²

The function $f(x) = x^2 - 8x + 19$ is written in standard form. What is the minimum value of $f(x)$?

Step-by-Step Explanation

Complete the square:

f(x) = (x² − 8x + 16) + 19 − 16 = (x − 4)² + 3

Vertex is (4, 3). Since a = 1 > 0, this is a minimum.

✅ Answer A: minimum value = $3$

Q11 Equilateral Triangle Height ★★☆

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KEY: equilateral triangle height = (side × √3) / 2 Perimeter = 3 × side → side = perimeter ÷ 3

The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is $k\sqrt{3}$ centimeters, where $k$ is a constant. What is the value of $k$?

Step-by-Step Explanation

side = 624 ÷ 3 = 208 cm

height = (208 × √3) / 2 = 104√3

So k = 104.

✅ Answer D: $k = 104$

Q12 Right Triangle Pythagorean Theorem ★☆☆

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KEY: a² + b² = c² (c = hypotenuse, always longest side) Memorize: 3-4-5, 5-12-13, 8-15-17 triangles

In a right triangle, the two legs have lengths 9 and 12. What is the length of the hypotenuse?

Step-by-Step Explanation

c² = 9² + 12² = 81 + 144 = 225 → c = 15

Note: √225 = 15, so B and C are equivalent — but 15 is the cleaner answer. Both B and C are accepted.

✅ Answer C: $15$

Q13 Circle Arc Length ★★☆

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KEY: arc length = (central angle / 360°) × 2πr sector area = (central angle / 360°) × πr²

A circle has radius 6. What is the arc length of a sector with a central angle of 120°?

Step-by-Step Explanation

arc length = (120/360) × 2π(6) = (1/3) × 12π = 4π

✅ Answer B: $4\pi$

Q14 Similar Triangles Proportions ★★☆

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KEY: similar triangles → corresponding sides are PROPORTIONAL cross multiply: a/b = c/d → ad = bc

Two similar triangles have corresponding sides in the ratio 3 : 5. If the perimeter of the smaller triangle is 36, what is the perimeter of the larger triangle?

Step-by-Step Explanation

Perimeters scale with the same ratio as side lengths:

36 / P = 3 / 5 → P = 36 × 5/3 = 60

✅ Answer C: $60$

Q15 Functions Composition ★★☆

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KEY: f(g(x)) means "plug g(x) into f" — work INSIDE OUT Always substitute completely before simplifying

Let $f(x) = 2x + 3$ and $g(x) = x^2 - 1$. What is $f(g(3))$?

Step-by-Step Explanation

g(3) = 3² − 1 = 9 − 1 = 8

f(g(3)) = f(8) = 2(8) + 3 = 16 + 3 = 19

✅ Answer B or C: $f(g(3)) = 19$

Q16 Word Problem Linear Model ★★☆

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KEY: "rate" = slope · "initial value" = y-intercept Build equation: total = (rate × time) + starting amount

A taxi charges a flat fee of $3.00 plus $1.50 per mile. Which equation represents the total cost $C$, in dollars, for a ride of $m$ miles?

Step-by-Step Explanation

Flat fee = y-intercept = 3.00. Rate per mile = slope = 1.50.

C = 1.50m + 3.00

✅ Answer C

Q17 Exponents Simplification ★★☆

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KEY: xᵃ × xᵇ = xᵃ⁺ᵇ · xᵃ ÷ xᵇ = xᵃ⁻ᵇ · (xᵃ)ᵇ = xᵃᵇ x⁰ = 1 · x⁻ⁿ = 1/xⁿ

Which expression is equivalent to $\dfrac{x^5 \cdot x^3}{x^4}$?

Step-by-Step Explanation

x⁵ × x³ = x⁸ → x⁸ / x⁴ = x⁸⁻⁴ = x⁴

✅ Answer C: $x^4$

Q18 Percentage Percent Change ★☆☆

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KEY: percent change = (new − old) / old × 100% "x% of y" = (x/100) × y

A shirt originally costs $40. It is on sale for 25% off. What is the sale price?

Step-by-Step Explanation

discount = 25% × $40 = 0.25 × 40 = $10

sale price = $40 − $10 = $30

✅ Answer C: $\$30$

Q19 Inequalities Number Line ★★☆

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KEY: multiply/divide by NEGATIVE → FLIP the inequality sign! −2x > 6 → x < −3 (sign flipped!)

Which value of $x$ satisfies $-3x + 2 > 11$?

Step-by-Step Explanation

−3x + 2 > 11 → −3x > 9 → x < −3 (flip sign!)

Check each option: x = 5 → −15+2 = −13 ✗ | x = 0 → 2 ✗ | x = −2 → 8 ✗ | x = −4 → 14 > 11 ✓

✅ Answer D: $x = -4$

Q20 Rational Equations Solving ★★★

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KEY: clear fractions → multiply BOTH sides by the LCD Always CHECK answers — denominators cannot equal zero!

What is the solution to $\dfrac{x}{x-2} = \dfrac{4}{x-2} + 1$?

Step-by-Step Explanation

Multiply both sides by (x − 2):

x = 4 + (x − 2) → x = 4 + x − 2 → x = x + 2 → 0 = 2

This is a contradiction — no solution. Also note: x = 2 would make the denominator zero, so it's excluded.

✅ Answer A: No solution