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Math: Advanced Math, 3 original questions

SAT Nonlinear Systems Practice: Questions and Answers

Find intersections of lines, parabolas and circles in three original SAT-style systems. Use substitution and verify both coordinates with complete explanations.

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The short answer

A solution of a system must satisfy every equation at the same time. When one equation already gives y, substitute that expression into the other equation or set the two expressions for y equal. Solve the resulting equation, then recover the matching y-value for each x-value. A line and a nonlinear curve may meet zero, one or multiple times. Count distinct coordinate pairs, not repeated algebraic factors, and check both original equations before keeping a solution.

Method at a glance

  1. 1Choose a substitution
  2. 2Solve the resulting equation
  3. 3Recover each coordinate pair
  4. 4Verify and count distinct pairs

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Choose a substitution

    Use an equation that isolates a variable. If both equations give y, equate their right sides to eliminate y.

  2. Step 2: Solve the resulting equation

    Factor or use another appropriate nonlinear method. Keep all real candidates that satisfy the restrictions.

  3. Step 3: Recover each coordinate pair

    Substitute each x-value into an original equation to find its matching y-value. Do not combine coordinates from different solutions.

  4. Step 4: Verify and count distinct pairs

    Check each pair in both equations. A repeated quadratic root gives one intersection rather than two different points.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

The system y = x² and y = x + 6 has two real solutions. What is the sum of their x-coordinates?

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Correct answer

Choice A: 1

Explanation

Set x² = x + 6, giving x² − x − 6 = 0. Factoring yields (x − 3)(x + 2) = 0, so x = 3 or −2. The corresponding pairs are (3, 9) and (−2, 4). Their x-coordinates sum to 1.

Why each choice is right or wrong

  1. Choice A

    Correct answer

    Correct: 3 + (−2) = 1.

  2. Choice B

    This adds the magnitudes of the roots instead of their signed values.

  3. Choice C

    This is the constant in the line equation, not the requested sum.

  4. Choice D

    This is one solution's y-coordinate, not the sum of x-coordinates.

Question 2 of 3

How many distinct real coordinate pairs solve y = x² + 2 and y = 4x − 2?

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Correct answer

Choice B: 1

Explanation

Equating the expressions gives x² + 2 = 4x − 2, so x² − 4x + 4 = 0. This is (x − 2)² = 0, with one distinct root x = 2. Both equations then give y = 6, so there is exactly one pair, (2, 6).

Why each choice is right or wrong

  1. Choice A

    The pair (2, 6) satisfies both equations, so there is a solution.

  2. Choice B

    Correct answer

    Correct: the repeated root describes one distinct pair.

  3. Choice C

    A quadratic can have two distinct roots, but this one repeats the same root.

  4. Choice D

    The coefficient 4 does not determine the number of solutions.

Question 3 of 3

A point (x, y) satisfies x² + y² = 25 and y = 3. If x is positive, what is x?

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Correct answer

Choice B: 4

Explanation

Substitute y = 3 into the first equation: x² + 9 = 25, so x² = 16. The possible x-values are 4 and −4. The positive condition selects 4, and 4² + 3² = 25 verifies the point.

Why each choice is right or wrong

  1. Choice A

    2² + 3² = 13, not 25.

  2. Choice B

    Correct answer

    Correct: the positive square root of 16 is 4.

  3. Choice C

    5² + 3² = 34, not 25.

  4. Choice D

    16 is x², not x.

Common mistakes

  • Keeping an x-value without finding its matching y-value. A system solution is a coordinate pair.

  • Counting a repeated root twice. Two identical factors can describe one distinct intersection.

  • Assuming every line meets every parabola or circle twice. The equations determine the number of intersections.

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Common questions

How is a nonlinear system different from a linear system?

At least one equation is nonlinear, so the intersection count can differ from the zero, one or infinitely many possibilities for two lines.

Can I solve by graphing?

A graph can show likely intersections. Verify the coordinates algebraically, especially when exact values or a tangent intersection matter.

Why do I need both coordinates?

Each solution represents one point satisfying the whole system. Finding an x-value alone does not identify its matching point.

Sources

  1. College Board: Advanced Mathsatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides