Lesson 2: Solving Systems of Linear Equations by 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

Back-substitution is the process of finding the first variable's value after solving for the second variable.
If you solved for $x$ first, substitute that value into either original equation to find $y$.
If you solved for $y$ first, substitute that value to find $x$.

Examples

Property

Many real-world situations can be modeled by a system of linear equations. The process involves:

1. Defining variables to represent the unknown quantities.
2. Writing two linear equations that describe the relationships between the variables.
3. Solving the system using the substitution method to find the solution.

Examples