Preparing for the SAT requires a solid understanding of algebra, as it’s a key part of the math section. In this post, we’ve compiled 25 essential SAT-style algebra questions, covering topics such as linear equations, inequalities, systems of equations, quadratic functions, and more. Challenge yourself with these questions and test your skills before exam day!
Part 1: Linear Equations and Inequalities
- Solve for x:
3x + 5 = 2x + 10 - If 4x – 7 = 2x + 9, what is the value of x?
- Solve the inequality:
5x – 3 ≥ 2x + 1 - Solve for y:
2(3y – 4) = 5y + 8 - What is the slope of the line that passes through the points (2, 3) and (6, 7)?
Part 2: Systems of Equations
- Solve the system of equations:
3x + 2y = 12
2x – y = 1 - If x + y = 5 and 2x – 3y = 4, what is the value of x and y?
- Solve the system of equations by substitution:
y = 2x + 1
3x + 2y = 14 - For the system of equations 4x + 3y = 9 and 2x – y = 5, find the values of x and y.
- Solve for x and y:
5x – 2y = 8
3x + 4y = -7
Part 3: Quadratic Functions
- Solve the quadratic equation:
x² – 5x + 6 = 0 - Solve for x:
2x² + 3x – 2 = 0 - The function f(x) = x² – 4x + 3 is graphed in the xy-plane. What are the coordinates of the vertex?
- What are the solutions to the quadratic equation x² – 4x = 12?
- If the sum of two numbers is 12 and their product is 32, what are the two numbers?
Part 4: Exponents and Polynomials
- Simplify:
(3x2)(2x3) - Simplify the expression:
4x3y2 / (2xy) - Factor the expression completely:
x2 – 9x + 14 - Simplify the expression:
(2x3 – 5x2 + 3x) – (x3 – 2x + 1) - Solve for x:
4x3 = 64
Part 5: Word Problems
- The length of a rectangle is 3 times its width. If the perimeter of the rectangle is 48 units, what are the dimensions of the rectangle?
- John is 5 years older than twice his sister’s age. If John is 23 years old, how old is his sister?
- A car rental company charges $25 per day plus $0.20 per mile driven. If Sarah rented a car and was charged $43, how many miles did she drive?
- The sum of three consecutive integers is 51. What are the integers?
- A train travels 300 miles at a constant speed. If it had gone 10 miles per hour faster, the trip would have taken 1 hour less. What is the speed of the train?
Next Steps
Algebra is a crucial component of the SAT math section. Be sure to practice questions like these to get a feel for the types of problems you’ll encounter on test day.
If you’re looking for solutions to these questions or more practice problems, find the solved solutions below:
SAT Algebra Practice Questions – Step-by-Step Solutions
Now, let’s walk through the step-by-step solutions for each question. Follow along and see how each problem is tackled.
Part 1: Linear Equations and Inequalities
- Solve for x:
3x + 5 = 2x + 10
• Subtract 2x from both sides: x + 5 = 10
• Subtract 5: x = 5 - Solve for x:
4x – 7 = 2x + 9
• Subtract 2x: 2x – 7 = 9
• Add 7: 2x = 16
• x = 8 - Solve the inequality:
5x – 3 ≥ 2x + 9
• Subtract 2x: 3x – 3 ≥ 9
• Add 3: 3x ≥ 12
• x ≥ 4 - Solve for y:
2(3y – 4) = 5y + 8
• Distribute: 6y – 8 = 5y + 8
• Subtract 5y: y – 8 = 8
• Add 8: y = 16 - Find the slope:
Through (2, 3) and (6, 7): slope = (7 – 3) / (6 – 2) = 4 / 4 = 1
Part 2: Systems of Equations
- Solve the system:
3x + 2y = 12
2x – y = 1
• From 2x – y = 1, y = 2x – 1
• Substitute: 3x + 2(2x – 1) = 12
3x + 4x – 2 = 12 → 7x = 14 → x = 2
• y = 2(2) – 1 = 3 - If x + y = 5 and 2x – 3y = 4:
• x = 5 – y
• Substitute: 2(5 – y) – 3y = 4
10 – 2y – 3y = 4 → 10 – 5y = 4
-5y = -6 → y = 6/5
• x = 5 – 6/5 = 19/5 - Solve by substitution:
y = 2x + 1
3x + 2y = 14
• Substitute: 3x + 2(2x + 1) = 14
3x + 4x + 2 = 14 → 7x = 12 → x = 12/7
• y = 2(12/7) + 1 = 31/7 - System of equations:
4x + 3y = 9
2x – y = 5
• y = 2x – 5
• 4x + 3(2x – 5) = 9
4x + 6x – 15 = 9 → 10x = 24 → x = 2.4
• y = 2(2.4) – 5 = 4.8 – 5 = -0.2 - Solve for x and y:
5x – 2y = 8
3x + 4y = -7
• Multiply first by 2: 10x – 4y = 16
• Multiply second by 5: 15x + 20y = -35
• Add: 25x = -19 → x = -19/25
• Substitute x into 5x – 2y = 8 to find y
Part 3: Quadratic Functions
- Solve x² – 5x + 6 = 0:
• Factor: (x – 2)(x – 3) = 0 → x = 2 or x = 3 - Solve 2x² + 3x – 2 = 0:
• Factor: (2x – 1)(x + 2) = 0 → x = 1/2 or x = -2 - Vertex of f(x) = x² – 4x + 3:
• x = -b/(2a) = 4/2 = 2
• f(2) = 4 – 8 + 3 = -1
• Vertex: (2, -1) - Solve x² – 4x = 12:
• x² – 4x – 12 = 0
• x = [4 ± √(16 + 48)] / 2 = [4 ± √64] / 2 → x = 6 or x = -2 - Two numbers sum to 12 and multiply to 32:
• x(12 – x) = 32 → x² – 12x + 32 = 0
• (x – 8)(x – 4) = 0 → x = 8 or x = 4
Part 4: Exponents and Polynomials
- Simplify (3x²)(2x³):
• 3 × 2 = 6; x² × x³ = x⁵ → 6x⁵ - Simplify 4x³y² / (2xy):
• 4/2 = 2; x³/x = x²; y²/y = y → 2x²y - Factor x² – 9x + 14:
• Numbers that multiply to 14 and add to -9: (-7) and (-2)
• (x – 7)(x – 2) - Simplify (2x³ – 5x² + 3x) – (x³ – 2x + 1):
• Distribute minus: 2x³ – 5x² + 3x – x³ + 2x – 1
• Combine like terms: x³ – 5x² + 5x – 1 - Solve 4x³ = 64:
• x³ = 16 → x = 2
Part 5: Word Problems
- Rectangle perimeter 48; length is 3 times width:
• Let width = w, length = 3w
• 48 = 2(3w + w) = 8w → w = 6
• Length = 18 - John 5 years older than twice his sister’s age, John is 23:
• 23 = 2x + 5 → x = 9 - Car rental: $25 + $0.20/mile = $43:
• 25 + 0.20m = 43 → 0.20m = 18 → m = 90 - Three consecutive integers sum to 51:
• x + (x+1) + (x+2)=51 → 3x = 48 → x=16
• Integers: 16, 17, 18 - Train 300 miles; 10 mph faster saves 1 hour:
• Let speed = x
• 300/x – 300/(x+10) = 1
• Multiply through by x(x+10): 300(x+10) – 300x = x(x+10)
• 3000 = x² + 10x → x² + 10x – 3000 = 0
• (x – 50)(x + 60) = 0 → x=50 mph
Conclusion
By working through these solutions, you should now have a clearer understanding of how to approach a variety of algebra problems on the SAT. Whether it’s solving systems of equations, factoring quadratics, or handling word problems, practice is key to improving your SAT math performance.
Stay tuned for more practice problems and tips for acing the SAT! Good luck with your SAT preparation!
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