math class 678 4-in-1 skill sheets
8.EE.C.8a 8th Grade Expressions & Equations

Systems by Graphing

Understand that solutions to a system of two linear equations correspond to the points where their graphs intersect.

How to explain it

At this standard, students solve a system of two linear equations by graphing both lines on the same coordinate plane, identify the intersection point as the ordered-pair solution, and recognize that parallel lines (same slope, different y-intercepts) indicate no solution.

The anchor students hold onto: Graph both lines on the same coordinate plane. The intersection point is the solution. If the lines are parallel (same slope, different y-intercepts), the system has no solution.

Students apply graphing intuition in Systems by Substitution and Systems by Elimination, where algebraic methods find the exact solution when graphing-by-eye is imprecise.

Worked examples

Example 1 Intersecting system
Solve: y = x + 1, x + 2y = 8.
Step 1Rewrite x + 2y = 8 as y = -1/2 x + 4.
Step 2Graph y = x + 1: y-int (0, 1), slope 1.
Step 3Graph y = -1/2 x + 4: y-int (0, 4), slope -1/2.
Step 4Lines meet at (2, 3).
AnswerSolution: (2, 3).
Example 2 Parallel system (no solution)
Identify: y = 2x+1 and y = 2x−3.
Step 1Both equations have slope 2.
Step 2y-intercepts: 1 and -3 (different).
Step 3Same slope + different intercepts = parallel.
Step 4Parallel lines never intersect.
AnswerNo solution.

Common mistakes

What students write Writing the intersection as (y, x) — reversing the coordinates.
The fix The horizontal axis is x, so the x-coordinate always comes first: write (x, y).
Try this A student solved the system y = 2x − 1 and y = −x + 5 by setting 2x − 1 = −x + 5 and got x = 2, y = 3. The student wrote the answer as (3, 2). Find and fix the student's error.
What students write Assuming every system must have a solution.
The fix Same slope + different y-intercepts = parallel lines = no solution. Check slopes first.

Teacher tip

Head off the two predictable errors before they happen. First: The horizontal axis is x, so the x-coordinate always comes first: write (x, y). Second: Same slope + different y-intercepts = parallel lines = no solution. Check slopes first.