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Understand Solutions as Intersection Points

Students graph two linear equations on the same coordinate plane and identify the solution of the system as the point of intersection. They recognize the three cases—one solution (lines intersect), no solution (parallel lines), and infinitely many solutions (same line)—and interpret each graphically.

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Understand Solutions as Intersection Points

Grade 8 · Systems of Linear Equations · question 1 of 3

0 of 20

20 rungs. A star at the top.

y = 2x + 1, y = −x + 28, x =

How it is taught

0:00 of 3:14

A solution of a system is a point on both lines.

If the two equations are the same line, every point on it works.

Read it: Where two lines cross

y = 6x − 2 and y = −2x + 14 are two lines. Where they cross, both give the same y, so 6x − 2 = −2x + 14. Then 8x = 16 and x = 2, and both lines give y = 10: the point (2, 10).

  1. Set the two right sides equal: where the lines cross, their y's are the same.
  2. Gather the x's on one side and the numbers on the other, then divide to find x.
  3. Put x back into either equation to find y.

A point is on a line when its x and y make the equation true.

The crossing is on both lines, so its y fits both equations at once: 6x − 2 and −2x + 14 are equal there.

At x = 2, 6 × 2 − 2 = 10 and −2 × 2 + 14 = 10, so (2, 10) is on both lines.

A system's solution is the point both lines pass through: set the y's equal, solve for x, then find y.

Stopping at x = 2 when the question asks for y.

x = 2 is half of the point. Put it back into either equation: y = 6 × 2 − 2 = 10.

y = 6x + 1, y = −3x + 46, x = ___ — 5. Set the right sides equal: 6x + 1 = −3x + 46, so 9x = 45 and x = 5.

y = x + 1 and y = −x + 3

Today we'll solve a system of two equations by finding where their graphs meet.

First, draw the coordinate plane — the x-axis goes across, the y-axis goes up.

Our first equation is y equals x plus one.

Plot it. This line slopes up and crosses the y-axis at one.

Our second equation is y equals negative x plus three.

Plot it too. This line slopes down and crosses the y-axis at three.

Look! The two lines cross at exactly one point.

That point is at x equals one and y equals two — written as one comma two.

Check the first equation: two equals one plus one. True!

Check the second: two equals negative one plus three. Also true!

Because the lines cross once, this system has exactly one solution.

Where do these two lines meet?y = x + 1 and y = −x + 3Oxyy = x + 1y = −x + 312(1, 2)2 = 1 + 1 ✓2 = −1 + 3 ✓One crossing → one solutionSOLUTION(1, 2)where both lines agree
Grade 8 · The solution of a system is where the lines cross

The solution is one comma two — the only point where both equations agree. Nice work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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