SAT Math: Nonlinear Systems of Equations Practice Questions
Substitute between equations, solve for intersection coordinates, and interpret how many real intersections exist. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. The system y = x^2 - 4 and y = 2x - 1 has two real solutions. What is the sum of their x-coordinates?
Explanation. Equate the two expressions for y: x^2 - 4 = 2x - 1, so x^2 - 2x - 3 = 0. Factoring gives (x - 3)(x + 1) = 0. The x-coordinates 3 and -1 sum to 2.
2. A point with positive x-coordinate satisfies x^2 + y^2 = 25 and y = 3. What is its x-coordinate?
Explanation. Substitute y = 3 into the first equation: x^2 + 9 = 25, so x^2 = 16. The two possibilities are 4 and -4; the positive coordinate is 4.
3. The graphs y = x^2 and y = 4x - 3 intersect twice. What is the larger y-coordinate of the intersection points?
Explanation. Set x^2 = 4x - 3 to get (x - 1)(x - 3) = 0. The corresponding y-values are 1^2 = 1 and 3^2 = 9. The larger y-coordinate is 9.
4. At the intersection of y = 2^x and y = 8, what is x?
Explanation. Solve 2^x = 8. Since 8 = 2^3, x = 3. The function 2^x is strictly increasing, so it reaches 8 only once.
5. The system y = x^2 + 2 and y = -x + 8 has two real solutions. What is the product of their x-coordinates?
Explanation. Substitution gives x^2 + x - 6 = 0, which factors as (x + 3)(x - 2) = 0. The x-coordinates are -3 and 2, whose product is -6.
6. How many real intersection points do y = x^2 and y = -x^2 + 18 have?
Explanation. Equating y-values gives 2x^2 = 18, or x^2 = 9. The two x-values, -3 and 3, both give y = 9. They produce two distinct intersection points.
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FAQ
Why can I set two expressions for y equal?
At an intersection, the same ordered pair satisfies both equations, so both expressions equal the same y-value.
Does each x-solution give a point?
Substitute each allowed x-value into an original equation to find y, then check the pair in the other equation. Distinct valid coordinate pairs are distinct intersections.