# Solving Systems of Equations

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```Solving Systems of
Equations
Substitution Method
Solving Systems by Substitution
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{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).
Example 1
EQ1: y  2 x
EQ 2 : y  x  5
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 + 5 becomes 2x = x + 5
Let’s solve for x !
2x  x  5
x x
x5
Plug into EQ1
EQ1: y = 2x becomes y = 2(5), therefore y = 10.
The solution point is (5, 10).
Example 2
EQ1: 2 x  y  5
EQ 2 : y  x  4
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
Simplify
Let’s solve for x !
2 x  ( x  4)  5
3x  4  5
4 4
3x  9
x3
Divide both sides by 3
Plug into EQ 2
EQ2: y = x - 4 becomes y = 3 – 4, therefore y = -1.
The solution point is (3, -1).
Consumer Economics Application
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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.
Write a System of Equations
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Write an equation for each option. Let y represent
the total amount paid and let x represent the
number of months.
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Equation 1: y = 30x + 50
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Equation 2: y = 40x
Solve the System
EQ1 : y  30 x  50 Both equations are solved for y, so we can
substitute either equation into the other.
EQ2 :
y  40 x Let’s choose to substitute 40x for y in EQ1.
Now y = 30x + 50 becomes 40x = 30x + 50
Let’s solve for x !
40 x  30 x  50
 30 x
 30 x
10 x  50
x5
Divide both sides by 10
Plug into EQ2
EQ2: y = 40x becomes y = 40(5), therefore y = 200.
The solution point is (5, 200).
Interpret the Results
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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 = 12
EQ1 = \$410
EQ2 = \$480
```