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

Question

A store is having a sale on jelly beans and trail mix. For 2 pounds of jelly beans and 5 pounds of trail mix, the total cost is $13. For 8 pounds of jelly beans and 3 pounds of trail mix, the total cost is $18. Find the cost for each pound of jelly beans and each pound of trail mix.

Answer 1

j = jellybeans

t = trail mix

2j + 5t = 13

8j + 3t = 18

To solve this, we want to isolate a variable, and solve for the other. Let's do that with each equation.

2j + 5t = 13

2j = 13 - 5t

j = (13 - 5t) / 2

8j + 3t = 18

8j = 18 - 3t

j = (18 - 3t) / 8

Now that we have isolated the variable j in each equation, we can set the two right-sides equal to each other, and solve for t.

(13 - 5t) / 2 = (18 - 3t) / 8

By multiplying both sides by 8, this becomes simpler to look at.

8 * (13 - 5t) / 2 = 18 - 3t

4 * (13 - 5t) = 18 - 3t

4 * 13 - 4 * 5t = 18 - 3t

52 - 20t = 18 - 3t

52 = 18 + 17t

34 = 17t

34 / 17 = t

t = 2

Now that we have solved for t, we can trivially plug it into one of the other equations to solve for j. Let's choose the first equation, as it doesn't matter which we choose.

t = 2

2j + 5t = 13

2j + 5 * 2 = 13

2j + 10 = 13

2j = 13 - 10

2j = 3

j = 1.5

So, each pound of jelly beans costs 1.5 dollars, or $1.50, and each pound of trail mix costs 2 dollars, or $2.

Answer 2

A store is having a sale on jelly beans and trail mix. For 2 pounds of jelly beans and 5 pounds of trail mix, the total cost is $13. For 8 pounds of jelly beans and 3 pounds of trail mix, the total cost is $18. Find the cost for each pound of jelly beans and each pound of trail mix.

Let J = No. of pounds of jelly beans

T = No. of pounds of trail mix

2J + 5T = 13

and

8J +3T = 18

Solve these two equations for T and J

8 J +20 T = 52 and

8J +3T = 18

gives 17T = 34

T= 2 and

2J +10 =13

2J = 3

J= 3/2 = 1.5

Jelly beans = 1.5 pounds and trail mix = 2 pounds

Answer 3

Let x = pounds of jelly beans and y = pounds of trail mix.

From the info above, we know:

2x + 5y = $13

and

8x + 3y = $18

We can multiply the first equation by 4 to get:

4 (2x + 5y = $13) --> 8x + 20y = $52

Then if we subtract the second equation from the first:

8x + 20y = $52 --> 8x + 20y = $52

-(8x + 3y = $18) --> -8x - 3y = - $18 (I carried the negative sign through the equation)

17y = $34

If we solve for y, we get:

17y = $34 --> y = $2

17 17

Now we can plug $2 in to either if the initial equations (i'll choose the first) and solve for x:

2x + 5($2) = $13 --> 2x + $10 = $13 --> 2x + $10 = $13 --> 2x = $3 --> 2x = $3 --> x = $1.5

-$10 -$10 2 2

Therefore, each pound of jelly beans costs $1.5 and each pound of trail mix costs $2