Solve this problem with a system of equations:
Suppose
you bought supplies for a party. You bought 3 rolls of streamers and 15 party hats
for $30. Later you bought 2 rolls of streamers and 4 party hats for $11. How
much did each roll of streamers cost? How much did each party hat cost?
This
step by step tutorial will show you how to solve a system of equations using
substitution.
First define the variable:
Let x= cost of streamers
Let y= cost of party hats
3x+15y=30 3
3 x+5y=10 -5y -5y x=-5y+10
Create a system of equations using the
given information:
3x+15y=30
(1)
2x+4y=11
(2)
Simplify the equations:
Divide both sides on equation (1)
by 3 and subtract 5y from both sides. Equation (1) is now transformed to
equation (3). Divide both sides on equation (2) by 2 and subtract 2y on both
sides. Equation (2) is now transformed to equation (4).
2x+4y=11 2
2 x+2y=11/2 -2y -2y x=-2y+11/2
x=-5y+10
(3)
x=-2y+ 11/2
(4)
Equate equations (3) and (4):
Since both equations equal x, they
also equal each other. Thus, we show this in the equation shown below. The
equation can be simplified to obtain y.
Subtract 2y on both sides and add 10 to both
sides.
-5y+10=-2y+11/2
+2y-10 +2y
-10
Divide both sides by -3.
-3y=-9/2
-3
-3
y=3/2 or $1.50
The next step is to substitute y into
equation (3) to obtain x:
You may substitute into equation (4) also.
x=-5y+10
x=-5(1.5)+10
x=-7.5+10
x=2.5 or $2.50
Your final answer is y=$1.50 and
x=$2.50.
Streamers
cost $2.50 and party hats cost $1.50.
Your last step is to explain how you
did it:
To solve this problem, I first created
a system of equations by making the total costs for each shopping trip equal
the items that were bought. I used variable to signify the unknown costs for
the different items. The coefficients for the variables showed how many of
those items were bought. I gave these equations the names equation 1 and
equation 2. Next, I simplified the
equations and gave these new equations the names 3 and 4. Since both
equations equaled x, I equated equations 3 and 4equations into one
equation. Then I isolated y on both sides and solved the equation. After that,
I substituted my solution for y into equation 3. Using substitution, I solved for x. My final answer was that
streamers cost $2.50 and party hats cost $1.50.
Another way to solve this:
You can also solve this equation by using
elimination. Use the same equations as the substitution problem. Multiply
equation 1 by 2 and multiply equation 2 by 3. This will make the coefficients
of x in both equations the same so that you can subtract to cancel or
eliminate the variable y out of the equations.
3x+15y=30
(1) Multiply by
2
2x+4y=11
(2) Multiply
by 3
Equations 1 and 2 become equations 3
and 4:
6x+30y= 60
(3)
6x+12y= 33
(4)
Subtract equation (4) from equation
(3) to get equation 5:
6x+30y= 60
-
6x+12y= 33
18y
= 27
18
18
y=3/2 or $1.50
Substitute y into equation (1):
3x+15y=30
3x+15(1.5) =30
3x+22.5 =30
Solve the equation:
3x+22.5=30
-22.5 -22.5
3x = 7.5
3
3
x=2.5 or $2.50
The
final answer is the same using substitution or elimination. Streamers cost
$2.50 and party hats cost $1.50; y=1.5 and x = 2.5.
By Sundipta R.
Period 7