Download Solving Linear Systems with Substitution

SOLVING LINEAR SYSTEMS
WITH SUBSTITUTION
by Sam Callahan
By now you’ve learned to solve systems of equations
using graphing and finding where the lines intersect:
y  x3
3x  y  7
y  x3
3x  y  7
The problem with solving by graphing though, is
evident when you look at graphs like the one below.
This solution (the blue point
where the lines intersect) isn’t
on a gridline and very
hard to accurately identify
Although graphing is simple and visual, it is really only
accurate enough to use with systems that have integer
answers.
Substitution is a way to solve systems of equations
analytically, or without graphing.
If we take the same system of equations:
y  x3
3x  y  7
We can make solving these equations possible by
working with one variable at a time. To do this, we
substitute one side of the equation in for the other
variable.
If we know that y = x+3,
we can plug in (x+3) in for y in the second equation to find out
what x is.
3x – y = 7
substitute (x+3) for y
3x – (x+3) = 7
distribute -1 through the parentheses
3x – x – 3 = 7
simplify
2x – 3 = 7
simplify
2x = 10
simplify
x=5
x=5
Now that we know what x is (5), we can plug 5 in for
x in either equation to find out what y is. I like to
use whichever equation has simpler numbers to work
with.
In this case, that equation is:
y  x3
If x = 5
y=x+3
y=5+3
y=8
x = 5 and y = 8, so our solution to the system
y  x3
3x  y  7
In (x, y) form is (5, 8)
Now let’s check our answer.
Checking your solution
You should check your answer using the equation that
you didn’t just solve.
For example, my last step was plugging in 5 for x into
y=x+3
I should check with the other equation, 3x – y = 7
Checking your solution
x=5
y=8
3x – y = 7
3(5) – (8) = 7
15 – 8 = 7
7=7
plug in your values for x and y
simplify
simplify
make sure your statement is true
We ended up with a true statement,
so our solution works!
Try this one…
4x – 12y = 20
3x + 9y = 45
Try this one…
4x – 12y = 20
3x + 9y = 45
Unlike the previous example, we aren’t given an equation
right away that says what x or y is equal to, so we have
to simplify one of these equations so that it reads
y=_____ or x=______
Choose one of the equations to simplify.
I’ll use 3x + 9y = 45
3x + 9y = 45
You can start with either variable, but I want to solve for
x first because I don’t want a fraction that would result
if I divided everything by 9.
3x + 9y = 45
3x = 45 – 9y
/3 /3 /3
x = 15 – 3y
subtract 9y to put it on the right side of the equation
divide everything by 3 so that x will be on its own
x = 15 – 3y
Now we have what x is equal to, so we can plug in
“15 – 3y” for x in the other equation.
4x – 12y = 20
4(15 – 3y) – 12y = 20
60 – 12y – 12y = 20
60 – 24y = 20
40 = 24y
plug in the expression for x
y = 5/3
I used a calculator for this step,
but you could also simplify this
fraction (40/24) by hand
distribute
simplify
simplify
simplify
y = 5/3
Plug this y-value into the other equation we found
(x = 15 – 3y) to find x.
x = 15 – 3y
x = 15 – 3(5/3)
x = 15 – 5
x = 10
plug in (5/3) for y
simplify
3 times (5/3) is 5
x = 10
So our solution is (10, 5/3)
Always check your solution!
y = 5/3
To review,
4x – 12y = 20
3x + 9y = 45
We simplified one of the equations
3x + 9y = 45  x = 15 – 3y

Plugged this “15 – 3y” in for x in the other equation
4x – 12y = 20
4(15 – 3y) – 12y = 20

Solved for y
y = 5/8

Plugged in this y-value into the other equation to find x
x = 15 – 3(5/3)
x = 10
