The lesson
This lesson teaches Solving Systems Algebraically. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
Substitution
When you study substitution, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
- 1Solve one equation for a variable.
- 2Substitute that expression into the other equation.
- 3Solve, then back-substitute to find the other variable.
Solve y = x + 1 and 2x + y = 7.
- 1Substitute y = x + 1: 2x + (x + 1) = 7.
- 2Combine: 3x + 1 = 7 → 3x = 6 → x = 2.
- 3Then y = 2 + 1 = 3. Solution: (2, 3).
Why this matters
Solving Systems Algebraically shows up constantly in use substitution or elimination for exact answers. It also connects to what you will see on homework, quizzes, and the next unit in this grade.
Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.
Common mistakes to avoid
Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.
Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.
Key ideas from this lesson
- Substitution
- Solve one equation for a variable.
- Substitute that expression into the other equation.
- Solve, then back-substitute to find the other variable.
Video walkthrough
Solving Systems by Substitution
Substitute one equation into the other to solve.
Watch on YouTubeSystems of Equations (Elimination)
Add or subtract equations to eliminate a variable.
Watch on YouTubePractice
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 yourselfSolve by substitution: y = x + 2 and 3x + y = 14.
Step-by-step solution
- 1Substitute: 3x + (x + 2) = 14.
- 24x + 2 = 14 → 4x = 12 → x = 3.
- 3y = 3 + 2 = 5; solution (3, 5).
Exercise 2
Try it yourselfSolve: y = 4x and x + y = 15.
Step-by-step solution
- 1x + 4x = 15 → 5x = 15.
- 2x = 3, then y = 4(3) = 12.
- 3Solution: (3, 12).
Exercise 3
Try it yourselfSolve by elimination: x + y = 10 and x − y = 2.
Step-by-step solution
- 1Add equations: 2x = 12, so x = 6.
- 2Substitute: 6 + y = 10 → y = 4.
- 3Solution: (6, 4).
Exercise 4
Try it yourselfSolve: 2x + y = 7 and 4x + 2y = 14.
Step-by-step solution
- 1Multiply the first equation by 2: 4x + 2y = 14.
- 2Both equations are identical.
- 3Infinitely many solutions.
Exercise 5
Try it yourselfSolve: 3x − 2y = 4 and 6x − 4y = 9.
Step-by-step solution
- 1Multiply the first by 2: 6x − 4y = 8.
- 2Second equation says 6x − 4y = 9.
- 38 = 9 is false; no solution.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Solving Systems Algebraically
Question 1 of 4
Solve the system y = 2x and x + y = 9.