Systems by Substitution

Grade 8 · algebra · 115 practice problems · read aloud

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Solving Systems of Equations by Substitution 🔁

This is a method to find where two lines on a graph intersect. It's useful because it gives you the one point (x, y) that makes both equations true at the same time. We use it to solve real-world problems, like figuring out when two plans cost the same amount.

Step-by-Step Guide

  1. Isolate a Variable: Look at both equations. Choose one and solve it for one variable (like y or x). Pick the one that looks easiest!
  2. Substitute: Take the expression you just found and substitute it into the other equation. This creates a new equation with only one variable.
  3. Solve: Solve this new equation for the remaining variable.
  4. Back-Substitute: Plug the value you just found back into one of the original equations to find the other variable.
  5. Check Your Solution: Plug both values into the original equations to make sure they work.

Worked Examples

Example 1: Solve the system: y = 2x and x + y = 12

  1. Step 1: The first equation is already solved for y: y = 2x
  2. Step 2: Substitute '2x' for 'y' in the second equation: x + (2x) = 12
  3. Step 3: Solve for x: 3x = 12 → x = 4
  4. Step 4: Back-substitute x=4 into y=2x: y = 2(4) → y = 8
  5. Step 5: Check: 8 = 2(4) ✔ and 4 + 8 = 12 ✔. Solution: (4, 8)

Example 2: Solve the system: 2x + y = 1 and x - 2y = 12

  1. Step 1: Solve the first equation for y: y = 1 - 2x
  2. Step 2: Substitute into the second equation: x - 2(1 - 2x) = 12
  3. Step 3: Solve for x: x - 2 + 4x = 12 → 5x = 14 → x = 14/5
  4. Step 4: Back-substitute: y = 1 - 2(14/5) = 1 - 28/5 = -23/5
  5. Solution: (14/5, -23/5)

Common Mistakes to Avoid 🚫

  • Substituting into the same equation: You must substitute into the other equation, not the one you just rearranged.
  • Distribution errors: Be careful with negative signs when substituting! For example, -2(1 - 2x) is -2 + 4x, not -2 - 4x.
  • Forgetting to find both variables: After you find x, you must go back and find y to have the complete solution point.

Tips & Tricks

  • Look for the "lonely" variable: If a variable has no coefficient (like y = 2x + 1 or x = 5), that's the easiest one to isolate first!
  • Label your steps: Write "Step 1: Solve for y" next to your work to stay organized.
  • Always check your answer! This catches small calculation errors and confirms you have the correct solution.

How to Practice

  • Start with systems where one variable is already isolated.
  • Move on to systems where you need to do one simple step to isolate a variable.
  • Create your own problems and solve them, or try to find the mistake in a purposely incorrect solution.
  • Use online practice websites for instant feedback on your answers.

Practice problems

6 of the 115, worked through step by step — try them before opening the answer.

1 2x + y = 7; y = 3x - 1

Hint: When one equation is already solved for a variable, substitute that expression into the other equation to create a single equation with one unknown.

Show the answer

Answer: x = 1.6, y = 3.8

  1. Substitute equation (2) into equation (1). Since equation (2) says y = 3x - 1, we can replace y in equation (1) with 3x - 1. So equation (1) becomes: 2x + (3x - 1) = 7
  2. Combine like terms. 2x + 3x - 1 = 7 5x - 1 = 7
  3. Add 1 to both sides. 5x - 1 + 1 = 7 + 1 5x = 8
  4. Divide both sides by 5. x = 8/5 x = 1.6
  5. Substitute x = 1.6 into equation (2) to find y. y = 3(1.6) - 1 y = 4.8 - 1 y = 3.8
  6. Check in equation (1). 2(1.6) + 3.8 = 3.2 + 3.8 = 7 ✓ Final answer: x = 1.6, y = 3.8

We are given the system of equations: 1) 2x + y = 7 2) y = 3x - 1

2 2x + 3y = 12, y = x - 1

Hint: When one equation is already solved for a variable, substitute that expression into the other equation to create a single equation with one variable.

Show the answer

Answer: x = 3, y = 2

  1. Substitute equation (2) into equation (1). Since y = x - 1, we replace y in equation (1) with (x - 1): 2x + 3(x - 1) = 12
  2. Simplify and solve for x. First, distribute the 3: 2x + 3x - 3 = 12 Combine like terms: 5x - 3 = 12 Add 3 to both sides: 5x = 15 Divide both sides by 5: x = 3
  3. Substitute x = 3 into equation (2) to find y. y = 3 - 1 y = 2
  4. Check the solution in equation (1). 2(3) + 3(2) = 6 + 6 = 12, which matches. Final answer: x = 3, y = 2

We are given the system of equations: 1) 2x + 3y = 12 2) y = x - 1

3 2x + 3y = 12; x = y + 1

Hint: When one variable is expressed in terms of another, substitute that expression into the other equation to create a single equation with one variable.

Show the answer

Answer: x = 3, y = 2

  1. Substitute the expression for x from equation 2 into equation 1. Since x = y + 1, replace x in equation 1 with (y + 1): 2(y + 1) + 3y = 12
  2. Expand and simplify. 2y + 2 + 3y = 12 (2y + 3y) + 2 = 12 5y + 2 = 12
  3. Isolate y. Subtract 2 from both sides: 5y = 12 - 2 5y = 10 Divide both sides by 5: y = 10 / 5 y = 2
  4. Find x using equation 2. x = y + 1 x = 2 + 1 x = 3
  5. Check the solution in equation 1. 2x + 3y = 2(3) + 3(2) = 6 + 6 = 12, which matches. Final answer: x = 3, y = 2

We are given the system of equations: 1) 2x + 3y = 12 2) x = y + 1

4 2x + 3y = 12 and y = x - 1

Hint: When one equation is already solved for a variable, substitute that expression into the other equation to create a single equation with one variable.

Show the answer

Answer: x = 3, y = 2

  1. Substitute y from equation 2 into equation 1. Since y = x - 1, replace y in equation 1 with (x - 1): 2x + 3(x - 1) = 12
  2. Simplify and solve for x. First, distribute 3 into (x - 1): 2x + 3x - 3 = 12 Combine like terms: 5x - 3 = 12 Add 3 to both sides: 5x = 15 Divide both sides by 5: x = 3
  3. Substitute x = 3 into equation 2 to find y. y = x - 1 y = 3 - 1 y = 2
  4. Check the solution in equation 1. Equation 1: 2x + 3y = 12 Substitute x = 3, y = 2: 2(3) + 3(2) = 6 + 6 = 12 This matches the given equation. Final answer: x = 3, y = 2

We have the system of equations: 1) 2x + 3y = 12 2) y = x - 1

5 2x + y = 11; y = 3x - 4; x = ?

Hint: When one equation is already solved for a variable, substitute that expression into the other equation where that variable appears.

Show the answer

Answer: 3

  1. Substitute y from equation 2 into equation 1. Replace y in equation 1 with (3x - 4): 2x + (3x - 4) = 11
  2. Combine like terms. 2x + 3x - 4 = 11 5x - 4 = 11
  3. Add 4 to both sides to isolate the term with x. 5x - 4 + 4 = 11 + 4 5x = 15
  4. Divide both sides by 5 to solve for x. 5x / 5 = 15 / 5 x = 3 Thus, the solution is x = 3. We can check by substituting x = 3 into equation 2: y = 3(3) - 4 = 9 - 4 = 5 Then equation 1: 2(3) + 5 = 6 + 5 = 11, which is correct. Final answer: x = 3

We are given the system of equations: 1) 2x + y = 11 2) y = 3x - 4 Since equation 2 gives y in terms of x, we can substitute this expression for y into equation 1.

6 3x - 2y = 7; y = 2x - 5; x = ?

Hint: Substitute the expression for y from the second equation into the first equation, then solve for x.

Show the answer

Answer: 3

  1. Start with the system: 3x - 2y = 7 and y = 2x - 5.
  2. Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 7.
  3. Distribute the -2: 3x - 4x + 10 = 7.
  4. Combine like terms: -x + 10 = 7.
  5. Subtract 10 from both sides: -x = -3.
  6. Multiply both sides by -1: x = 3. The value of x is 3.
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