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Mom's Basement University

Systems of Equations: Substitution Method

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn the substitution method for solving systems of two linear equations, including when to use it and how to handle special cases.

The Concept

Section titled “The Concept”

The substitution method solves a system by:

  1. Solving one equation for one variable (usually the easiest one, isolated or with coefficient 1)
  2. Substituting that expression into the other equation
  3. Solving the resulting one-variable equation
  4. Substituting the value back into either original equation to find the second variable
  5. Writing the solution as (x, y) and checking in both equations

Example system:

{y=3x−22x+y=7\begin{cases} y = 3x - 2 \\ 2x + y = 7 \end{cases}

  1. First equation already solves for y: y = 3x − 2
  2. Substitute into second: 2x + (3x − 2) = 7
  3. 5x − 2 = 7 → 5x = 9 → x = 9/5 = 1.8
  4. y = 3(1.8) − 2 = 5.4 − 2 = 3.4

Solution: (9/5, 17/5) or (1.8, 3.4)

Special cases:

  • No solution: substitution leads to contradiction (e.g., 0 = 5)
  • Infinite solutions: substitution leads to identity (e.g., 0 = 0)

Worked Example

Section titled “Worked Example”

Solve:

{3x+2y=13x=y+1\begin{cases} 3x + 2y = 13 \\ x = y + 1 \end{cases}

  1. Second equation already solved for x: x = y + 1
  2. Substitute into first: 3(y + 1) + 2y = 13
  3. 3y + 3 + 2y = 13 → 5y + 3 = 13 → 5y = 10 → y = 2
  4. x = 2 + 1 = 3

Solution: (3, 2)

Check:

  • 3(3) + 2(2) = 9 + 4 = 13 ✓
  • x = 3, y + 1 = 2 + 1 = 3 ✓

Both true.

Another (no solution case):

{2x+3y=54x+6y=12\begin{cases} 2x + 3y = 5 \\ 4x + 6y = 12 \end{cases}

  1. Solve first for x: 2x = 5 − 3y → x = (5 − 3y)/2
  2. Substitute: 4((5 − 3y)/2) + 6y = 12
  3. 2(5 − 3y) + 6y = 12 → 10 − 6y + 6y = 12 → 10 = 12 (contradiction)

No solution (parallel lines).

Real-World Application

Section titled “Real-World Application”

Substitution is great for systems where one variable is easy to isolate:

  • Two phone plans: Plan A: 30 + 0.10t = total, Plan B: 45 + 0.05t = total. Set equal: 30 + 0.10t = 45 + 0.05t → solve for t (texts when equal)
  • Two jobs: Job A: 20h + 50 bonus, Job B: 18h + 120 bonus. 20h + 50 = 18h + 120 → substitution finds h (hours when pay matches)
  • Mixture: Blend 5 dollars per pound coffee with 8 dollars per pound to get 6.50 dollars per pound. Set up system and substitute to find amounts.

This method helps compare options, find balance points, or mix solutions.

Quiz

Section titled “Quiz”
Solve using substitution: y = 2x + 1 and 3x + y = 11.
Solve: x + y = 10 and y = 3x − 2.
Two plans: A: 40 + 0.05t, B: 25 + 0.10t. When equal?
Solve: 2x + y = 10 and x = y − 1.
Solve: $y = x + 3$ and $2x + y = 12$.