Solving linear equations using the substitution method involves isolating one variable in one equation and substituting that expression into another equation. This results in a single equation with a single variable, which can then be easily solved. Here's a step-by-step guide:
Steps to Solve Linear Equations by Substitution
Here's how to solve linear equations by substitution:
- Simplify the equations: If necessary, simplify the given equations by expanding any parentheses or combining like terms. This makes the equations easier to work with.
- Solve for one variable: Choose one of the equations and solve it for one of the variables (either x or y). This means isolating that variable on one side of the equation. The goal is to express one variable in terms of the other.
- Substitute: Substitute the expression you found in step 2 into the other equation. This will give you a new equation that contains only one variable.
- Solve the new equation: Solve the equation you obtained in step 3 for the remaining variable. This can be done using basic arithmetic operations.
Example
Let's consider the following system of linear equations:
- Equation 1: x + y = 5
- Equation 2: 2x - y = 1
Step 1: The equations are already simplified.
Step 2: Solve Equation 1 for x:
x = 5 - y
Step 3: Substitute this expression for x into Equation 2:
2(5 - y) - y = 1
Step 4: Solve for y:
10 - 2y - y = 1
10 - 3y = 1
-3y = -9
y = 3
Now that you have the value of y, you can substitute it back into either Equation 1 or Equation 2 (or the expression x = 5 - y) to solve for x.
Using x = 5 - y:
x = 5 - 3
x = 2
Therefore, the solution to the system of equations is x = 2 and y = 3.
