Simultaneous equations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Where two lines meet

Mathematics · Algebra · ages 13-14
Name ______________________   Date ____________

  1. Line A is y = 2x and Line B is y = x + 3. They cross at (3, 6). What is true about the point (3, 6)?

    • It only solves y = 2x
    • It only solves y = x + 3
    • It solves both equations

  2. Lines y = x + 2 and y = 6 - x cross at (2, 4). The point (0, 2), which lies on y = x + 2, also solves the system.

    Circle one:   True   False

  3. A graph places the crossing between x = 1 and x = 2, and between y = 3 and y = 4. Which pair is a reasonable estimate?

    • (3.5, 1.5)
    • (1.5, 3.5)
    • (2.5, 4.5)
    • (1.5, 2.5)

  4. Two lines meet at (4, -1). What does this point mean for their two equations?

    • A point that lies on only one of the two lines
    • A point that makes both equations true at once
    • A point that gives the larger intercept of the two lines

  5. Test x = 2, 3, 4 and 5 in y = 3x - 5 and y = x + 3. At which x-value do the two equations give the same y?

    Answer: ______________

  6. Nadia thinks (1, 4) solves both y = x + 3 and y = 2x + 2. She checks the first equation: 4 = 1 + 3, true. What should she do next?

    • Stop after the first equation matches
    • Substitute the point into the other equation too
    • Choose a new point before checking the second equation

  7. Lena tests the pair (2, 5) in y = 3x - 1 and y = x + 4. What does she find?

    • It fits y = x + 4 but not y = 3x - 1
    • It fits both equations, so it solves the system
    • It fits y = 3x - 1 but not y = x + 4
    • It fits neither of the two equations

  8. A graph suggests y = x + 4 and y = 3x - 2 cross at x = 3. Substitute x = 3 into y = x + 4 to find y.

    Answer: ______________

  9. A graph shows a crossing near (1, 4). Which check supports that estimate?

    • Both equations give y = 4 when x = 1
    • The equations give different y-values when x = 1
    • Only the first equation is checked when x = 1

  10. Omar checks a proposed solution (3, 5) in y = x + 2: 5 = 3 + 2 is true. He checks it in y = 2x - 1: 5 = 2(3) - 1 = 5, also true. What can Omar conclude?

    • (3, 5) solves both equations
    • (3, 5) solves only the first equation
    • He needs to graph the lines to be sure
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Where two lines meet W1-mt_mqgu72aCMz-s1

  1. It solves both equations · Since (3, 6) sits on both lines, it satisfies both equations at once.
  2. False · (0, 2) fits y = x + 2, since 0 + 2 = 2, but 6 - 0 = 6, not 2. Only the crossing (2, 4) lies on both lines.
  3. (1.5, 3.5) · The x-coordinate comes first and lies between 1 and 2. The y-coordinate comes second and lies between 3 and 4, so (1.5, 3.5) fits.
  4. A point that makes both equations true at once · Every point on a line fits that line's equation. A point on both lines makes both equations true, so (4, -1) is a solution of the pair.
  5. 4 · At x = 4 both equations give 7: 3(4) - 5 = 7 and 4 + 3 = 7. At x = 2, 3 and 5 the outputs differ, so the lines cross at (4, 7).
  6. Substitute the point into the other equation too · A real solution must work in both equations, so she still needs to check the second one.
  7. It fits y = 3x - 1 but not y = x + 4 · 3(2) - 1 = 5, so the pair fits y = 3x - 1. But 2 + 4 = 6, not 5, so it fails y = x + 4 and does not solve the system.
  8. 7 · y = x + 4, so y = 3 + 4 = 7. The other line agrees, since 3(3) - 2 = 7.
  9. Both equations give y = 4 when x = 1 · At the crossing, both equations must give the same y for the same x. If both give y = 4 at x = 1, the outputs match each other and the estimate, so the check supports (1, 4).
  10. (3, 5) solves both equations · Since (3, 5) makes both equations true, it is the shared solution. No graph is needed to confirm it.
Worksheet · LightMySky