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OhioMathsSyllabus dot point

How does the graph of a system of linear equations show its solution, and how do you read the solution as the point where the two lines cross?

Solve systems of two linear equations by graphing, reading the solution as the intersection point, and connect the graph to the algebraic outcome (Ohio A-REI.6, A-REI.11).

An Ohio Algebra I answer on solving linear systems by graphing (A-REI.6): graphing each line, reading the intersection as the solution, and what parallel and identical lines mean for the number of solutions.

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  1. What this topic is asking
  2. Graphing each line
  3. What the three pictures look like
  4. When graphing is exact and when it is not
  5. How Ohio examines this topic
  6. Why the intersection is the solution
  7. Try this

What this topic is asking

Ohio standard A-REI.6 also asks you to solve a system by graphing: graph both lines and read the intersection point as the solution. The graph makes the meaning of a solution visible, it is the single (x,y)(x, y) that lies on both lines at once. Graphing items live in the Expressions and Equations and Functions categories and use the online coordinate grid.

Graphing each line

Put each equation in a form you can graph, usually slope-intercept y=mx+by = mx + b, then plot.

What the three pictures look like

The algebraic outcome and the graph always agree.

So before graphing, a quick look at the slopes predicts the picture: equal slopes mean parallel or identical, unequal slopes mean exactly one crossing.

When graphing is exact and when it is not

Graphing is reliable when the intersection lands on a clear lattice point (whole-number coordinates). When the solution has fractions, like (3,3.5)(3, 3.5), the graph only estimates it, and the exact value comes from substitution or elimination. On the test, a graphing item is usually designed so the crossing is a clean point you can click, while messier systems appear as equation-response items solved algebraically.

How Ohio examines this topic

  • Graphing. Plot two lines, then click the intersection point.
  • Multiple choice and multiple-select. Match a system to its graph, or decide the number of solutions from the slopes and intercepts.
  • Equation response. Confirm the intersection point you read.

Why the intersection is the solution

The graph of a linear equation is the set of all points that make that equation true. So the graph of the system is the set of points that make both equations true, which is exactly where the two line-graphs overlap. For two distinct non-parallel lines, that overlap is a single point, the intersection, which is why "solve the system" and "find where the lines cross" are the same task. This is also why parallel lines give no solution (no shared point) and identical lines give infinitely many (every point is shared). Seeing the solution this way ties the algebra in the substitution and elimination methods to a picture you can reason about.

Try this

Q1. The lines y=xy = x and y=−x+4y = -x + 4 cross at what point? [2 points]

  • Cue. x=−x+4x = -x + 4, so 2x=42x = 4, x=2x = 2, y=2y = 2; (2,2)(2, 2).

Q2. A system graphs as two lines that lie exactly on top of each other. How many solutions? [1 point]

  • Cue. Identical lines, so infinitely many.

Exam-style practice questions

Practice questions written in the style of ODEW exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Ohio Algebra I EOC (style)2 marksGraphing item. The lines y=x+1y = x + 1 and y=−2x+7y = -2x + 7 are graphed. State the solution of the system as an ordered pair.
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The solution is (2,3)(2, 3).

The solution of a system is the point where the two lines cross. Setting the expressions equal, x+1=−2x+7x + 1 = -2x + 7, gives 3x=63x = 6, so x=2x = 2; then y=2+1=3y = 2 + 1 = 3. On a graphing item you would click the lattice point (2,3)(2, 3). Checking both: y=x+1y = x + 1 gives 33, and y=−2x+7y = -2x + 7 gives −4+7=3-4 + 7 = 3, so the point lies on both lines. The intersection is the visual meaning of "solves both equations."

Ohio Algebra I EOC (style)1 marksMultiple choice. Two lines in a system have the same slope but different y-intercepts. How many solutions does the system have? (A) none (B) one (C) two (D) infinitely many
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The correct answer is (A).

Lines with the same slope are parallel. Different yy-intercepts mean they are not the same line, so they never cross, and a system whose graphs never meet has no solution. If the slopes differed, the lines would cross once (one solution); if both the slope and the intercept matched, the lines would coincide and there would be infinitely many solutions.

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