Substitution Method for Linear Equations- Practice Problems

What Is the Substitution Method?

The substitution method is a way to solve systems of linear equations. You solve one equation for one variable, then plug that expression into the other equation. That's it. Nothing fancy.

It works best when one variable has a coefficient of 1 or -1. If both variables have ugly coefficients, the elimination method is usually faster.

How to Use Substitution (The Actual Steps)

Here's the process:

• Solve one equation for one variable in terms of the others
• Substitute that expression into the remaining equation
• Solve the resulting single-variable equation
• Plug your answer back into one of the original equations to find the other variable
• Write your solution as an ordered pair (x, y)

Let's walk through a real example.

Example 1: Basic Substitution

Problem: Solve the system
y = 2x + 3
3x + y = 11

Step 1: The first equation already has y isolated. Use it.

Step 2: Substitute 2x + 3 for y in the second equation.

3x + (2x + 3) = 11

Step 3: Solve for x.

5x + 3 = 11
5x = 8
x = 8/5 = 1.6

Step 4: Plug x back into y = 2x + 3.

y = 2(1.6) + 3
y = 3.2 + 3
y = 6.2

Solution: (1.6, 6.2)

Example 2: Neither Variable Is Isolated

Problem: Solve the system
2x + y = 7
x - 3y = -2

Step 1: Solve the first equation for y.

y = 7 - 2x

Step 2: Substitute into the second equation.

x - 3(7 - 2x) = -2
x - 21 + 6x = -2
7x - 21 = -2
7x = 19
x = 19/7

Step 3: Find y.

y = 7 - 2(19/7)
y = 7 - 38/7
y = 49/7 - 38/7
y = 11/7

Solution: (19/7, 11/7)

Practice Problems

Try these before checking the solutions. No peeking.

Problem 1

Solve the system:
y = x - 4
2x + y = 10

Solution:
Substitute x - 4 for y:
2x + (x - 4) = 10
3x - 4 = 10
3x = 14
x = 14/3

y = 14/3 - 4 = 14/3 - 12/3 = 2/3

Answer: (14/3, 2/3)

Problem 2

Solve the system:
3x - 2y = 8
x + 4y = -6

Solution:
Solve x + 4y = -6 for x:
x = -6 - 4y

Substitute into 3x - 2y = 8:
3(-6 - 4y) - 2y = 8
-18 - 12y - 2y = 8
-18 - 14y = 8
-14y = 26
y = -26/14 = -13/7

Find x:
x = -6 - 4(-13/7)
x = -6 + 52/7
x = -42/7 + 52/7
x = 10/7

Answer: (10/7, -13/7)

Problem 3

Solve the system:
5x + y = 3
3x - 2y = -1

Solution:
Solve for y: y = 3 - 5x

Substitute:
3x - 2(3 - 5x) = -1
3x - 6 + 10x = -1
13x - 6 = -1
13x = 5
x = 5/13

Find y:
y = 3 - 5(5/13)
y = 3 - 25/13
y = 39/13 - 25/13
y = 14/13

Answer: (5/13, 14/13)

Problem 4

Solve the system:
y = -2x + 1
4x + 2y = 2

Solution:
Substitute -2x + 1 for y:
4x + 2(-2x + 1) = 2
4x - 4x + 2 = 2
2 = 2

This is true. The system has infinitely many solutions—the equations are equivalent.

Answer: Infinitely many solutions (dependent system)

Problem 5

Solve the system:
y = 3x - 5
6x - 2y = 10

Solution:
Substitute 3x - 5 for y:
6x - 2(3x - 5) = 10
6x - 6x + 10 = 10
10 = 10

Same result. Infinitely many solutions again.

Answer: Infinitely many solutions (dependent system)

Problem 6

Solve the system:
x - 2y = 3
2x - 4y = 5

Solution:
Solve x - 2y = 3 for x: x = 3 + 2y

Substitute:
2(3 + 2y) - 4y = 5
6 + 4y - 4y = 5
6 = 5

Contradiction. No solution exists.

Answer: No solution (inconsistent system)

Substitution vs Elimination: Which to Use?

Here's a quick comparison:

Most textbooks let you choose. Pick whatever creates less work.

Common Mistakes to Avoid

• Forgetting parentheses when substituting. Write out the full expression before simplifying.
• Solving for the wrong variable. Make sure you substitute into the other equation, not the one you just used.
• Arithmetic errors when distributing negatives. 3(2x - 5) = 6x - 15, not 6x + 15.
• Dropping negative signs when moving terms. Subtracting -2y means adding 2y.

Quick Reference Checklist

• ✓ Is one variable isolated (or easy to isolate)?
• ✓ Did you substitute the entire expression?
• ✓ Did you simplify correctly?
• ✓ Did you back-substitute to find the other variable?
• ✓ Did you write the answer as an ordered pair?

That's substitution. Practice the problems above until you can do them without checking the solutions. The process is mechanical—once you see the pattern, you won't overthink it.