Unit 5 · Topic 2

Solving Systems Algebraically

Overview

Use substitution or elimination for exact answers.

Topic 2 of 2~24 min
All 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.

Substitution

  1. 1Solve one equation for a variable.
  2. 2Substitute that expression into the other equation.
  3. 3Solve, then back-substitute to find the other variable.
Worked example

Solve y = x + 1 and 2x + y = 7.

  1. 1Substitute y = x + 1: 2x + (x + 1) = 7.
  2. 2Combine: 3x + 1 = 7 → 3x = 6 → x = 2.
  3. 3Then y = 2 + 1 = 3. Solution: (2, 3).

Why this matters

Use substitution or elimination for exact answers.

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. Substitution
  2. Solve one equation for a variable.
  3. Substitute that expression into the other equation.
  4. Solve, then back-substitute to find the other variable.

Video walkthroughs

Khan Academy

Solving Systems by Substitution

Substitute one equation into the other to solve.

Watch on YouTube
Khan Academy

Systems of Equations (Elimination)

Add or subtract equations to eliminate a variable.

Watch on YouTube
Other

Solving a System by Graphing

Graph both lines and read the intersection point.

Watch on YouTube
Khan Academy

Graphs of Linear Equations

Review graphing before comparing algebraic solution methods.

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 substitution: y = x + 2 and 3x + y = 14.

Step-by-step solution

  1. 1Substitute: 3x + (x + 2) = 14.
  2. 24x + 2 = 14 → 4x = 12 → x = 3.
  3. 3y = 3 + 2 = 5; solution (3, 5).

Exercise 2

Try it yourself

Solve: y = 4x and x + y = 15.

Step-by-step solution

  1. 1x + 4x = 15 → 5x = 15.
  2. 2x = 3, then y = 4(3) = 12.
  3. 3Solution: (3, 12).

Exercise 3

Try it yourself

Solve by elimination: x + y = 10 and x − y = 2.

Step-by-step solution

  1. 1Add equations: 2x = 12, so x = 6.
  2. 2Substitute: 6 + y = 10 → y = 4.
  3. 3Solution: (6, 4).

Exercise 4

Try it yourself

Solve: 2x + y = 7 and 4x + 2y = 14.

Step-by-step solution

  1. 1Multiply the first equation by 2: 4x + 2y = 14.
  2. 2Both equations are identical.
  3. 3Infinitely many solutions.

Exercise 5

Try it yourself

Solve: 3x − 2y = 4 and 6x − 4y = 9.

Step-by-step solution

  1. 1Multiply the first by 2: 6x − 4y = 8.
  2. 2Second equation says 6x − 4y = 9.
  3. 38 = 9 is false; no solution.

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

Khan Academy

Systems with substitution & elimination

Open resource

Quick check

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

Solving Systems Algebraically

Question 1 of 4

Medium

Solve the system y = 2x and x + y = 9.

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