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The graph shows a system of linear
equations. As you can see, without the
use of technology, determining the
solution from the graph is not easy. You
can use the substitution method to find
an exact solution. In substitution, you
solve one equation for one variable and
then substitute this expression into the
other equation.
You can also solve systems of equations with the elimination
method. With elimination, you get rid of one of the variables by
adding or subtracting equations. You may have to multiply one or
both equations by a number to create variable terms that can be
eliminated.
Reading Math
The elimination method is sometimes called the addition
method or linear combination.
Ex 1: Use substitution to solve the system of equations.
y= x–1
x+y=7
Step 1 Solve one equation for one variable. The first equation is
already solved for y: y = x – 1.
Step 2 Substitute the expression into the other equation.
x+y=7
2x = 8
x + (x – 1) = 7
x=4
2x – 1 = 7
Step 3 Substitute the x-value into one of the original equations to
solve for y.
y=x–1
y = (4) – 1
y=3
The solution is the ordered pair (4, 3).
Ex 2: Use elimination to solve the system of equations.
3x + 5y = –16
2x + 3y = –9
Step 1 To eliminate x, multiply both sides of the first equation by 2
and both sides of the second equation by –3.
2(3x + 5y) = 2(–16)
6x + 10y = –32
–3(2x + 3y) = –3(–9)
–6x – 9y = 27
Add the equations.
y = –5
Step 2 Substitute the y-value into one of the original equations to
solve for x.
3x + 5(–5) = –16
3x – 25 = –16
3x = 9
x =3
The solution for the system is (3, –5).
Ex 4: A veterinarian needs 60 pounds of dog food that is 15%
protein. He will combine a beef mix that is 18% protein
with a bacon mix that is 9% protein. How many pounds
of each does he need to make the 15% protein
mixture?
Write one equation based on the amount of dog food: x+ y = 60
Write another equation based on the
amount of protein:
Solve the system.
0.18x+ 0.09y =0.15(60)
x + y = 60
0.18x +0.09y = 9
x + y = 60
y = 60 – x
0.18x + 0.09(60 – x) = 9
0.18x + 5.4 – 0.09x = 9
0.09x = 3.6
x = 40
Substitute x into one of the original equations to solve for y.
40 + y = 60
The mixture will contain 40 lb of the beef
y = 20
mix and 20 lb of the bacon mix.