Substitution Method (Systems of Equations)

Definition

The substitution method is a technique used to solve a system of linear equations by solving one equation for one variable and then substituting that expression into the other equation. This allows you to find the values of both variables.

Steps for the Substitution Method:

1. Solve one equation for one variable: Choose one of the equations and solve for one of the variables (usually the variable with the smaller coefficient).

2. Substitute the expression into the other equation: Substitute the expression you found in Step 1 into the other equation.

3. Solve for the remaining variable: After substituting, solve the resulting equation to find the value of the remaining variable.

4. Substitute back: Once you have the value of the second variable, substitute it back into the equation you used in Step 1 to find the value of the first variable.

Example:

Consider the system of equations:

1. 3x + y = 11

2. x - y = 1

Step 1: Solve one equation for one variable

Solve the second equation for x:

x - y = 1 → x = y + 1

Step 2: Substitute the expression for x into the first equation

Now substitute x = y + 1 into the first equation 3x + y = 11:

3(y + 1) + y = 11

Step 3: Solve for y

Distribute the 3 in the equation:

3y + 3 + y = 11

Simplify:

4y + 3 = 11

Subtract 3 from both sides:

4y = 8

Solve for y:

y = 8/4 = 2

Step 4: Substitute y = 2 back into the expression for x

Now substitute y = 2 into the equation x = y + 1:

x = 2 + 1 = 3

Final Solution:

The solution to the system of equations is x = 3 and y = 2, or the point of intersection is (3, 2).

Why it's Called "Substitution":

The name "substitution" comes from the fact that you substitute the expression for one variable into the other equation to eliminate that variable and solve for the remaining one. This method is particularly useful when one of the equations is easy to solve for one variable.