1. Solving Systems of Equations
Substitution Method

2. Solving Systems by Substitution
{Step 1} Solve for one variable in at least one equation if necessary
{Step 2} Substitute the resulting expression into the other equation.
{Step 3} Solve the equation equation to obtain the first variable
{Step 4} Substitute that value into one of the original equations and solve for the other variable
{Step 5} Write the values from Step 3 and Step 4 as an ordered pair, (x, y).
{Step 6} Check your solution

3. Example 1
Both equations are solved for y, so we can substitute either equation into the other. Let’s choose to substitute 2x for y in EQ2.
Now y = x becomes 2x = x + 5
Let’s solve for x !
Plug into EQ1
EQ1: y = 2x becomes y = 2(5), therefore y = 10.
The solution point is (5, 10).

4. Example 2
The second equation is solved for y, so we can substitute x – 4 into y for EQ1.
Now 2x + y = 5 becomes 2x + (x – 4) = 5
Let’s solve for x !
Simplify
Divide both sides by 3
Plug into EQ 2
EQ2: y = x becomes y = 3 – 4, therefore y = -1.
The solution point is (3, -1).

5. Consumer Economics Application
One high-speed internet service provider has a $50 setup fee, and costs $30 per month. Another provider has no setup fee and costs $40 per month.
After how many months will both providers have the same cost?
What will the cost be?
If you plan to cancel in one year, which is the cheaper provider? Explain.

6. Write a System of Equations
Write an equation for each option. Let y represent the total amount paid and let x represent the number of months.
Equation 1: y = 30x + 50
Equation 2: y = 40x

7. Solve the System
Both equations are solved for y, so we can substitute either equation into the other. Let’s choose to substitute 40x for y in EQ1.
Now y = 30x becomes 40x = 30x + 50
Let’s solve for x !
Divide both sides by 10
Plug into EQ2
EQ2: y = 40x becomes y = 40(5), therefore y = 200.
The solution point is (5, 200).

8. Interpret the Results
Since x represents the number of months and x = 5, this means that at 5 months both plans will be have equal cost.
Since y represents the total cost and y = 200, this means that after 5 months both plans will have cost $500
Which plan is cheaper after 1 year? Plug 12 in for x and compare the results.
When x = EQ1 = $ EQ2 = $480