Systems of equations test

One equation solved for x, then two solved together for the point they share.

Systems of equations test

The answer key prints on a separate page.

What's on this test

A single equation has one unknown and one answer; a system has two of each, and its solution is the one pair that satisfies both at once — the point where the two lines cross. That shift is what the paper tests, which is why the two-step section is here at all: solving a system comes down to solving one clean equation once a variable has been removed, and a child shaky on section 3 will stall in section 1 for a reason that is not about systems.

How the paper is put together

Recognizing whether an equation is even linear sits in the middle because it decides what tool is allowed. A system of straight lines meets at a single point; curves do not have to. Sorting the linear from the non-linear before solving is the check that stops a child from applying elimination to something it was never going to work on.

Math test

Name: Date:

1. Systems of equationsSolve each pair by adding the two equations. Give x and y.6 questions

12x + 3y = 239x − 3y = −12x = y =

23x + 6y = −184x − 6y = 18x = y =

32x + 12y = 2612x − 12y = −12x = y =

45x + 9y = 452x − 9y = −45x = y =

53x + 11y = −34−2x − 11y = 41x = y =

64x + 10y = −34−11x − 10y = −29x = y =

2. Linear or nonlinearIs each equation linear or nonlinear? Circle one.5 questions

7y = x/9 + 23LinearNonlinear

8y = 5x²LinearNonlinear

9y = 14x + 38LinearNonlinear

10y = 39/xLinearNonlinear

11y = 9x² + 5LinearNonlinear

3. Solving equationsSolve each equation for x.5 questions

123x + 28 = 40

133x + 25 = 46

142x + 43 = 45

154x − 29 = 19

165x − 20 = 5

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Answer key

Systems of equations

1. 2x + 3y = 23, 9x − 3y = −12 → (1, 7)2. 3x + 6y = −18, 4x − 6y = 18 → (0, −3)3. 2x + 12y = 26, 12x − 12y = −12 → (1, 2)4. 5x + 9y = 45, 2x − 9y = −45 → (0, 5)5. 3x + 11y = −34, −2x − 11y = 41 → (7, −5)6. 4x + 10y = −34, −11x − 10y = −29 → (9, −7)

Linear or nonlinear

7. y = x/9 + 23: Linear — dividing by 9 is multiplying by a number, so the steps in y stay equal8. y = 5x²: Nonlinear — x is squared, so the steps in y grow as x grows9. y = 14x + 38: Linear — x is only multiplied by a number and added to, so equal steps in x give equal steps in y10. y = 39/x: Nonlinear — x is underneath, so doubling x halves y instead of adding the same amount again11. y = 9x² + 5: Nonlinear — x is squared, so the steps in y grow as x grows

Solving equations

12. 3x + 28 = 40 → x = 413. 3x + 25 = 46 → x = 714. 2x + 43 = 45 → x = 115. 4x − 29 = 19 → x = 1216. 5x − 20 = 5 → x = 5

Systems of equations test

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