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Graphing Systems of Equations Practice Problems

Graph both lines on one grid and read off where they cross. Use slope-intercept form or the two intercepts to draw each line, then confirm the crossing point algebraically.

  1. 1.

    Graph and . Where do the lines cross?

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    The lines start at and with slopes 1 and −1 and cross at : ✓, ✓.

  2. 2.

    Graph and . Where do the lines cross?

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    From up 2, right 1, and from down 1, right 1: the lines meet at .

  3. 3.

    Graph and using intercepts. Where do they cross?

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    passes through and ; through and . They cross at .

  4. 4.

    Graph and . Where do the lines cross?

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    is horizontal. Setting gives , so they meet at .

  5. 5.

    Graph and . Where do the lines cross?

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    Intercepts and , slopes 3 and −2. They cross at : ✓ and ✓.

  6. 6.

    Graph and using intercepts. Where do they cross?

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    has intercepts and ; has and . They cross at .

  7. 7.

    Graph and . How many solutions does the system have? Answer 0, 1 or "infinitely many".

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    Both lines have slope 2 with different intercepts, so they are parallel and never cross.

  8. 8.

    Taxi A charges $3 plus $2 per mile; Taxi B charges $5 plus $1.50 per mile. Graph both fares. At what point do they cross?

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    and cross where : , .

  9. 9.

    A line has -intercept and slope ; another has -intercept and slope . Where do they intersect?

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    The lines are and . gives , .

  10. 10.

    One line passes through with slope 1; another passes through and . Where do they intersect?

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    The lines are and . Setting them equal gives , .

  11. 11.

    The lines and cross between grid points. What is the exact -coordinate of the intersection? Give a fraction.

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    gives , so . The graph can only estimate this; algebra gives it exactly.

  12. 12.

    Estimate from a graph, then find exactly: where do and intersect? Give the exact pair using fractions.

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    gives , so and (about ).
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Graphing Systems of Equations Practice Problems

These problems follow the graphing systems of equations lesson. Sketch each pair of lines on graph paper, like a solving systems by graphing worksheet, then check your crossing point in both equations.

What the graph tells you

LinesSolutions
cross onceone
parallelnone
the same lineinfinitely many

Common mistakes to watch for

  • Reading the intersection loosely. Graphs give estimates; substitute the point into both equations to confirm it.
  • Calling parallel lines a solution. Parallel lines never meet, so the system has no solution.
  • Missing identical lines. If both equations describe the same line, every point on it is a solution.
  • Graphing from the wrong form. Rewrite each equation as before plotting, so slope and intercept are clear.

Frequently Asked Questions

How do I graph each line quickly?+

In slope-intercept form, plot the y-intercept and step out the slope. In standard form, plot the x- and y-intercepts and join them.

What if the lines cross between grid points?+

Estimate from the graph, then set the two expressions equal to get the exact values.

How do I type my answer?+

Type an ordered pair such as (2, 3); fractions like (4/3, 5/3) are fine.

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