Solving a word problem using a system of linear equations of the form Ax + By = C

Question

A store is having a sale on chocolate chips and walnuts. For 8 pounds of chocolate chips and 4 pounds of walnuts, the total cost is $33. For 3 pounds of chocolate chips and 2 pounds of walnuts, the total cost is $13. Find the cost for each pound of chocolate chips and each pound of walnuts.

Answer 1

Let "x" be the cost of one pound of chocolate chips. Let "y" be the cost of one pound of walnuts.

8x + 4y = 33

3x + 2y = 13

Multiply the bottom equation by 2 and subtract the first equation from the second one.

8x + 4y = 33

-(6x + 4y = 26)

2x = 7

x = 7/2

Use x to find y

3(7/2) + 2y = 13

2y = 5/2

y = 5/4

Answer 2

The question is asking for price for pound of chocolate chips and walnuts.

Step 1 define the variables.

x= price per pound of chocolate chips

y= price per pound of walnuts

Step 2 determine equations from the information given.

8x + 4y= 33

3x + 2y= 13

Step 3 Solve system of Equations using Elimination Method

Using elimination method you want to add or subtract equations and eliminate 1 variable. Since combining our equations will not eliminate a variable we have to multiply the second equation by a coefficient to be able to eliminate a variable. We will use (-2)

8x + 4y = 33

-2* ( 3x + 2y) =13* (-2)........-6x + 2y = -26

8x + 4y = 33

+ -6x - 4y = -26

2x + 0y = 7 add equations

2x = 7 solve for x

x=7/2 = 3.5

Next substitute x into one the equations and solve for y

3x + 2y = 13

(3)(3.5]) +2y =13

10.5 + 2y = 13

2y=13-10.5

y= 2.5/2 = 1.25

Solution x, price of chocolate Chip is $3.5/lb

y, price of walnuts is $1.25/lb

check your work

you can substitute these values for x and y into the equation 8x + 4y = 33

(8*3.5) + (4*1.25) = 33

28 + 5 = 33 solution checks

There is a second method called substitution method that can be used instead of the elimination method.

for substitution method you solve one variable in terms of the other variable.

8x + 4y= 33

3x + 2y= 13

3x+2y =13

2y= 13-3x

y= (13-3x)/2

y= 6.5-1.5x substitute y into the first equation

8x + 4(6.5 - 1.5x) =33

8x + 26 - 6x =33

2x = 7

x=3.5

substitute x into equation and solve for y

y= 6.5-1.5x

y = 6.5- (1.5*3.5)

y = 6.5 - 5.25

y = 1.25

Solution x, price of chocolate Chip is $3.5/lb

y, price of walnuts is $1.25/lb