SOLVING SYSTEM OF EQUATIONS BY SUBSTITUTION

KEY CONCEPTS There are various methods to solving a

system of equations. Two methods include the substitution method and the elimination method.

The substitution method involves solving one of the equations for one of the variables and substituting that into the other equation.

Solutions to systems are written as an ordered pair, (x,y). This is where the lines would cross if graphed.

KEY CONCEPTS CONTINUED If the resulting solution is a true

statement, such as 9 = 9, then the system has an infinite number of solutions. This is where the lines would coincide if graphed.

If the result is an untrue statement, such as 4 = 9, then the system has no solutions. This is where lines would be parallel if graphed.

Check your answer by substituting the x and y values back into the original equations. If the answer is correct, the equations will result in true statements.

STEPS FOR SUBSTITUTION METHOD Step 1: Solve one of the equations for

one of the variables in terms of the other variable.

Step 2: Substitute, or replace, the resulting expression into the other equation.

Step 3: Solve the equation for the second variable.

Step 4: Substitute the found value into either of the original equations to find the value of the other variable.

EXAMPLE 1:

Step 1: Solve one of the equations for one of the variables in terms of the other variable. It doesn’t matter which equation you choose, nor does

it matter which variable you solve for.

Let’s solve for the variable y.

Isolate y by subtracting x from both sides.

Step 2: Substitute, or replace, into the other equation, . It helps to place parentheses around the expression you are

substituting.

Step 3: Solve the equation for the second variable.

Second equation of the system. Substitute for y.

Distribute the negative over Simplify.Add 2 to both sides.

Divide both sides by 2.

Step 4: Substitute the found value, (), into either of the original equations to find the value of the other variable.

The solution to the system of equations is (). If graphed, the lines would cross at ().

First equation of the system.

Substitute for .

Simplify.

Subtract from both sides.

EXAMPLE 2:

Step 1: Solve one of the equations for one of the variables in terms of the other variable. It doesn’t matter which equation you choose, nor does

it matter which variable you solve for.

Let’s solve for the variable y.

Isolate by adding to both sides.

Step 2: Substitute, or replace, into the other equation, . It helps to place parentheses around the expression you are

substituting.

Step 3: Solve the equation for the second variable.

Second equation of the system. Substitute for .

Distribute the 4 through Simplify.Add 12 to both sides.

Divide both sides by 5.

Step 4: Substitute the found value, (), into either of the original equations to find the value of the other variable.

The solution to the system of equations is (). If graphed, the lines would cross at ().

First equation of the system.

Substitute for y.

Simplify.

Add 8 to both sides.

YOU TRY!1) 2)

REAL-WORLD APPLICATIONSExample 1: A shopper purchased 4 tables and 2 chairs for $200 and 2 tables and 7 chairs for $400. What is the cost of each table and each chair?

Set up a system of equations:

4 𝑥+2 𝑦=200

EXAMPLE 1 CONTINUED Solve for x and y:

REAL-WORLD APPLICATIONSExample 2: If the length of the rectangle is twice the width, and the perimeter of the rectangle is 30 cm, what is the length and width of the rectangle? Set up a system of equations:

EXAMPLE 2 CONTINUED Solve for and :