HighSchoolQ

➕ Algebra 1 · Systems of Equations

Solving Systems by Graphing: notes and common mistakes

Study notes

Key ideas

  • A system of two linear equations asks for an ordered pair (x, y) that makes both equations true. That pair is the solution of the system.
  • Each line is the set of solutions of one equation. A point on both lines is a solution of both, so the solution is where the lines cross.
  • To solve by graphing: write each equation as y = mx + b, graph both lines on the same axes, read the crossing, and check it in both equations.
  • For non-vertical lines: different slopes give one solution. The same slope with different y-intercepts (parallel lines) gives no solution. The same slope and the same y-intercept (the same line) gives infinitely many solutions.
  • Graphing is exact only when the crossing is a point you can read exactly and the check works. Otherwise it is an estimate.

Steps

  1. Write each equation as y = mx + b (solve for y if you need to).
  2. Graph both lines on the same axes with one scale: plot (0, b), then use the slope.
  3. Find where the lines cross and read (x, y).
  4. Check the point in both original equations.
  5. Write the solution as an ordered pair, or say no solution or infinitely many solutions.

Formulas

  • Solution of a system: An ordered pair (x, y) that makes both equations true
  • One solution: m1 ≠ m2 (non-vertical lines with different slopes): the lines cross exactly once
  • No solution: m1 = m2 and b1 ≠ b2: parallel lines
  • Infinitely many solutions: m1 = m2 and b1 = b2: the same line
  • x-coordinate of the crossing: Solve m1x + b1 = m2x + b2 for x, then find y

Watch out

  • Test the point in both equations. A point on only one line fits only one equation.
  • Give both coordinates as an ordered pair, x first: (1, 3), not just x = 1.
  • The same slope does not by itself mean no solution. Compare the y-intercepts: different means parallel (no solution), the same means one line (infinitely many).
  • If the lines do not meet on your grid, compare the slopes before you say “no solution.” Lines with different slopes cross somewhere, maybe off the edge of your graph.
  • A crossing between grid lines is only an estimate. Check it, and use algebra for the exact answer.
  • When you solve an equation for y, divide every term, and keep the signs right when you divide by a negative number.

Why it works

  • Why is the crossing point the solution? Each line is the set of solutions of one equation. A point on both lines is in both sets, so it makes both equations true. And any solution of the system must be on both lines.
  • Why do lines with different slopes cross exactly once? Write them as y = m1x + b1 and y = m2x + b2. The vertical gap between them is (m1 − m2)x + (b1 − b2). If the slopes differ, this gap changes by the same nonzero amount for each 1 step in x, so it is zero at exactly one value of x. If the slopes are equal, the gap is the same everywhere: never zero (parallel) or always zero (the same line).
  • Why does the x-coordinate of the crossing solve m1x + b1 = m2x + b2? At the crossing both lines have the same y, so the two expressions for y are equal there.

Common mistakes

✗ MistakeThe solution of a system is the x-value, like x = 2.
✓ InsteadThe solution is an ordered pair (x, y) that makes both equations true. Both numbers are needed.
✗ MistakeIf a point is on one of the lines, it is a solution of the system.
✓ InsteadIt must be on both lines, so it must make both equations true. Check it in both.
✗ MistakeTwo lines with the same slope have no solution.
✓ InsteadThat is true only if the y-intercepts differ (parallel lines). With the same slope and the same y-intercept, the lines are the same, and there are infinitely many solutions.
✗ MistakeIf the lines do not cross on my graph, the system has no solution.
✓ InsteadLines with different slopes always cross somewhere, even if that is off the edge of the graph. Compare the slopes to decide.
✗ MistakeA graph always gives the exact answer, so I do not need to check.
✓ InsteadA reading from a graph can be slightly off, and a crossing between grid lines can only be estimated. Check it in both equations.

Teach it back

Explain to a friend how to solve the system y = −3x + 10 and y = x − 2 by graphing, and how to be sure the answer is right.

Explaining it out loud shows what you really know. Did you…

  • I graphed the first line from (0, 10) with slope −3 (down 3, right 1) and the second from (0, −2) with slope 1 (up 1, right 1).
  • I read the crossing point (3, 1), with x first and then y.
  • I checked it in both equations: −3(3) + 10 = 1 and 3 − 2 = 1.
  • I said that the slopes (−3 and 1) are different, so there is exactly one solution, and that parallel lines would mean no solution while the same line would mean infinitely many.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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