Solving Systems by Substitution

Substitution replaces a variable with an expression known to have the same value, turning two equations into one equation in one variable.

When substitution is natural

Look for an equation where $x$ or $y$ is already isolated, or has coefficient $1$ or $-1$. If $y=2x+3$, every occurrence of $y$ in the other equation may be replaced by $2x+3$.

The replacement keeps all solutions because the first equation states that the two quantities are equal.

A reliable method

1. Isolate one variable in either equation.
2. Substitute the complete expression into the other equation.
3. Solve the resulting one-variable equation.
4. Back-substitute to find the other variable.
5. Check the ordered pair in both originals.

Use brackets around the substituted expression, especially after a negative coefficient.

Worked example

Check: $7=2(3)+1$ and $3(3)+7=16$.

Isolating first

For $x+y=9$ and $2x-y=3$, the first equation gives $y=9-x$. Substitute into the second:

Distribute the negative: $2x-9+x=3$, so $3x=12$, $x=4$, and $y=5$.

Fractions can be strategic

Avoiding fractions is helpful but not mandatory. If $2x+4y=7$, isolating $x$ gives $x=(7-4y)/2$. Elimination may be cleaner. Method choice is part of efficient problem solving, not a test of loyalty to one algorithm.

No solution

Substitution may remove the variable and produce a contradiction. For $y=2x+1$ and $y=2x-4$, substituting gives $2x+1=2x-4$, then $1=-4$. The lines are parallel and the system has no solution.

Infinitely many solutions

If substitution produces an identity such as $0=0$, both equations describe the same line. Every point satisfying either equation solves the system.

For example, $y=3x-2$ and $2y=6x-4$ are equivalent.

Modelling example

One quantity is often naturally expressed in terms of another. If an adult ticket costs $5$ more than a student ticket and together they cost $23$, let student price be $s$ and adult price $a=s+5$. Then $s+(s+5)=23$, so $s=9$ and $a=14$.

Common mistakes

Substituting into the same equation used to isolate. Use the other equation to create new information.

Dropping brackets. Replace the variable with the entire expression.

Stopping after one variable. A two-variable system needs an ordered pair.

Misreading a contradiction or identity. Interpret it as none or infinitely many solutions.

Quick self-check

• Which variable is cheapest to isolate?
• Did I substitute into the other equation?
• Are brackets protecting the replacement?
• Did I back-substitute and check both equations?

Related topics

• Systems of Linear Equations
• Linear Equation
• Solving Systems by Graphing
• Solving Systems by Elimination