6-2 Substitution

Property

The substitution method is an algebraic technique for solving a system of equations by replacing one variable with an equivalent expression.
To solve a system by substitution, follow these four steps:

1. Isolate: Choose one equation and isolate one variable (make its coefficient 1 or -1 to avoid fractions).
2. Substitute: Plug that isolated expression into the OTHER equation. This creates a new equation with only one variable.
3. Solve: Solve this new one-variable equation.
4. Back-Substitute: Plug the value you just found back into the isolated equation from Step 1 to find the second variable's value.

Examples

• Isolating First: Solve $3x + y = 10$ and $2x + y = 5$.

The easiest variable to isolate is $y$ in the first equation. Subtract $3x$ to get $y = 10 - 3x$.

• Full Substitution: Solve $y = 3x$ and $x + y = 12$.

Since $y$ is already isolated, substitute $3x$ for $y$ in the second equation: $x + 3x = 12$.
Solve: $4x = 12$, so $x = 3$.
Back-substitute: $y = 3(3) = 9$. The solution is $(3, 9)$.

• Multi-Step: Solve $a - 2b = 5$ and $3a + b = 8$.

Isolate $a$ in the first equation: $a = 2b + 5$.
Substitute into the second: $3(2b + 5) + b = 8$.
Solve: $6b + 15 + b = 8 \rightarrow 7b = -7 \rightarrow b = -1$.
Back-substitute: $a = 2(-1) + 5 = 3$. The solution is $(3, -1)$.

Explanation

Think of the substitution method as a perfectly legal mathematical swap. Because an equation tells you two things are perfectly equal (like $y = 3x$), you can completely remove the $y$ from the other equation and drop the $3x$ in its place. Choosing to isolate a variable that already has a coefficient of 1 or -1 is a pro-tip—it saves time and prevents you from having to do algebra with messy fractions!

Property

When substituting an expression that contains multiple terms, you must always enclose the entire expression in parentheses.
If there is a coefficient or a negative sign in front of the variable you are replacing, the distributive property must be applied to every term inside those parentheses:

Examples

• Sign Error with Subtraction: Solve using $y = 4x + 11$ and $3x - y = -8$.

Substitute with parentheses: $3x - (4x + 11) = -8$.
Correct distribution: $3x - 4x - 11 = -8$.
(Common Mistake: Writing $3x - 4x + 11 = -8$ by forgetting to distribute the negative to the 11).

• Coefficient Distribution: Solve using $x = 2y - 3$ substituted into $4x + 5y = 6$.

Substitute with parentheses: $4(2y - 3) + 5y = 6$.
Correct distribution: $8y - 12 + 5y = 6$.
(Common Mistake: Writing $8y - 3 + 5y = 6$ by only multiplying the first term).

• Negative Coefficient: Solve using $x = -3y + 7$ substituted into $-2x + y = 1$.

Substitute with parentheses: $-2(-3y + 7) + y = 1$.
Correct distribution: $6y - 14 + y = 1$.

Explanation

Parentheses act as a protective container for your substituted expression. When you drop a new expression into an equation, any number or negative sign sitting outside must be multiplied by every single piece inside the container. Writing the parentheses explicitly on your paper before doing any mental math is the single most reliable way to prevent heartbreaking sign and distribution errors.

Property

When using substitution, the variables will sometimes completely cancel each other out. The resulting numerical statement determines the number of solutions and the geometric classification of the system:

• Identity (True Statement): If solving results in a true statement like $0 = 0$ or $5 = 5$, the system has Infinitely Many Solutions. The equations are dependent (they represent the exact same coincident line).
• Contradiction (False Statement): If solving results in a false statement like $0 = -10$ or $2 = -1$, the system has No Solution. The equations are inconsistent (they represent parallel lines).

Examples

• Infinite Solutions (Identity): Solve $y = 2x - 3$ and $4x - 2y = 6$.

Substitute $y$: $4x - 2(2x - 3) = 6$.
Distribute: $4x - 4x + 6 = 6$.
Simplify: $6 = 6$. This is a true statement, so there are infinitely many solutions.

• No Solution (Contradiction): Solve $y = 5x + 2$ and $y = 5x - 1$.

Substitute $y$: $5x + 2 = 5x - 1$.
Subtract $5x$ from both sides: $2 = -1$. This is a false statement, so there is no solution.

Explanation

If your variables vanish during substitution, the algebra is trying to tell you something about the geometry of the lines! A true statement like $6 = 6$ means the two equations are actually identical twins disguised as different math problems; they overlap perfectly. A false statement like $2 = -1$ means you have reached a mathematical dead-end because the lines are parallel and will literally never cross.