Solving Systems of Equations by Substitution

When you have a system of two linear equations, your goal is to find the values of the variables (usually $x$ and $y$) that make both equations true at the same time. One of the most direct ways to find this solution is by using the substitution method.

How the Substitution Method Works

The main idea behind substitution is to reduce a system of two equations with two variables into a single equation with just one variable. Here are the standard steps:

1. Isolate: Pick one equation and solve it for one of the variables (e.g., get $y = \dots$ or $x = \dots$).
2. Substitute: Plug that expression into the other equation. This gives you a single equation with only one variable.
3. Solve: Solve this new equation to find the value of the first variable.
4. Plug back in: Substitute the value you just found back into the isolated equation from Step 1 to find the second variable.

Example 1

Let's solve the following system: $y = 2x + 1$ $3x + y = 11$

Step 1: Isolate The first equation is already solved for $y$ ($y = 2x + 1$).

Step 2: Substitute Replace the $y$ in the second equation with $(2x + 1)$: $3x + (2x + 1) = 11$

Step 3: Solve Now, solve for $x$: $5x + 1 = 11$ $5x = 10$ $x = 2$

Step 4: Plug back in Substitute $x = 2$ back into the first equation to find $y$: $y = 2(2) + 1$ $y = 4 + 1$ $y = 5$

The solution to the system is the coordinate pair $(2, 5)$.

Example 2

Solve this system: $x + y = 10$ $2x - y = 5$

Step 1: Isolate Let's isolate $x$ in the first equation by subtracting $y$ from both sides: $x = 10 - y$

Step 2: Substitute Plug $(10 - y)$ in place of $x$ in the second equation: $2(10 - y) - y = 5$

Step 3: Solve Distribute and solve for $y$: $20 - 2y - y = 5$ $20 - 3y = 5$ $-3y = -15$ $y = 5$

Step 4: Plug back in Substitute $y = 5$ back into our isolated equation from Step 1: $x = 10 - 5$ $x = 5$

The solution to this system is $(5, 5)$.