Unit 5 · Topic 1

Systems by Graphing

Overview

The solution is the point where the two lines cross.

Topic 1 of 2~22 min
Next: Solving Systems AlgebraicallyAll units
Unit overview

The lesson

Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.

Graph both lines

  1. 1Write each equation in slope-intercept form if needed.
  2. 2Graph both lines on the same grid.
  3. 3The intersection point is the solution.
Worked example

Where do y = x + 1 and y = -x + 3 intersect?

  1. 1Graph both lines.
  2. 2They cross at (1, 2).
  3. 3Check: 1 + 1 = 2 ✓ and -1 + 3 = 2 ✓.

Why this matters

The solution is the point where the two lines cross.

These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.

Common mistakes to avoid

Rushing to the answer without writing steps. Clear work helps you catch mistakes and shows your reasoning.

Re-reading the question too quickly. Underline what you are solving for before you start calculating.

Key ideas from this lesson

  1. Graph both lines
  2. Write each equation in slope-intercept form if needed.
  3. Graph both lines on the same grid.
  4. The intersection point is the solution.

Video walkthroughs

Other

Solving a System by Graphing

Graph both lines and read the intersection point.

Watch on YouTube
Khan Academy

Graphs of Linear Equations

Graph two lines to prepare for finding their intersection.

Watch on YouTube
Khan Academy

Graphing a Line in Slope-Intercept Form

Plot each equation quickly using slope-intercept form.

Watch on YouTube
Khan Academy

Slope-Intercept Form

Rewrite equations so they are easy to graph.

Watch on YouTube

Practice

For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.

Exercise 1

Try it yourself

Solve by graphing: y = x and y = 2. What is the solution?

Step-by-step solution

  1. 1y = 2 is a horizontal line at height 2.
  2. 2y = x crosses it when x = 2, y = 2.
  3. 3Solution: (2, 2).

Exercise 2

Try it yourself

The lines y = 3x + 1 and y = 3x − 5 are graphed. How many solutions does the system have?

Step-by-step solution

  1. 1Both lines have slope 3 but different y-intercepts.
  2. 2Parallel lines never intersect.
  3. 3No solution.

Exercise 3

Try it yourself

Graph y = −x + 4 and y = x. Where do they intersect?

Step-by-step solution

  1. 1−x + 4 = x.
  2. 2Add x: 4 = 2x, so x = 2.
  3. 3y = 2; intersection (2, 2).

Exercise 4

Try it yourself

y = 2x + 1 and y = 2x + 1 are graphed together. How many solutions?

Step-by-step solution

  1. 1Both equations describe the same line.
  2. 2Every point on the line satisfies both equations.
  3. 3Infinitely many solutions.

Exercise 5

Try it yourself

A system’s graph shows lines crossing at (−1, 5). What is the solution?

Step-by-step solution

  1. 1The intersection point satisfies both equations.
  2. 2x = −1 and y = 5.
  3. 3Solution: (−1, 5).

Re-read the lesson section if you get stuck, then try again without peeking at the solution.

Khan Academy

Systems by graphing

Open resource

Quick check

Answer all questions. Retake the quiz until you feel confident before moving on.

Systems by Graphing

Question 1 of 4

Easy

The solution to a system of two lines is found where the lines do what?

Next →

1-on-1 tutoring

Stuck on something specific?

Adam & Alan can walk through it live. Free consultation call for new clients.

Book a tutoring session