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 almonds. For 3 pounds of jelly beans and 8 pounds of almonds, the total cost is $18. For 5 pounds of jelly beans and 2 pounds of almonds, the total cost is $13. 

Find the cost for each pound of jelly beans and each pound of almonds.

Answer 1

This question involves solving a system of linear equations.

First, let's name some variables to define our system.

We'll refer to the cost per pound of jelly beans as the variable "J," and the cost per pound of almonds as the variable "A." Both variables, J & A, will have units of ($/lb)

Now, we can use these variables to write our system of equations.

For the first equation, we're told that 3 pounds of jelly beans and 8 pounds of almonds yields a total cost of $18.

So, we can write that 3*J+8*A = $18 (Equation 1)

For the second equation, we're told that 5 pounds of jelly beans and 2 pounds of almonds yields a total cost of $13.

So, we can write that 5*J+2*A = $13 (Equation 2)

Equation 1 and 2 each contain two unknown variables, J & A. However, because we have two equations relating each variable to one another, we can solve the system by making a substitution.

Let's begin with equation 1. We need to solve for one of the variables "in terms of" or "as a function of" the other. It doesn't matter which one you choose; your initial choice will affect which of the two variables you solve for first.

Using equation 1, 3*J+8*A = $18, let's solve for the variable "A".

First, subtract the term "3*J" over to the right side of the equation. This gives us 8*A = 18 - 3*J.

Now, we can divide both sides of the equation by 8 to solve for the variable A.

This will give us A = (18/8) - (3/8)*J. We'll call this (Equation 3).

Because we've now isolated A in terms of J, we can substitute equation 3 into equation 2 which will allow us to solve for J.

Recall, Equation 2 5*J+2*A = $13 and Equation 3 A = (18/8) - (3/8)*J

Using our function for A in equation 3, we will substitute this equation into the variable A present in Equation 2.

This gives us: 5*J + 2*[(18/8) - (3/8)*J] = 13

Distributing or multiplying the 2 into the bracketed term, we get 5*J + (36/8) - (6/8)*J = 13.

We now need to isolate J to one side of the equation. To do this, we will subtract the (36/8) term to the right side of the equation.

This gives us 5*J - (6/8)*J = 13 - (36/8). In order to solve for J, we first need to find a common denominator on the left side of the equation so that we can combine those terms. To do this, multiply by 8/8.

(5*8/8)*J - (6/8)*J = (13*8/8) - (36/8) -> (40/8)*J - (6/8)*J = (104/8) - (36/8)

Now that we have a common denominator, simply the above equation by combining like terms.

This will yield the equation (34/8)*J = (68/8) or in decimals, 4.25*J = 8.5.

Divide both sides of the equation by 4.25 to solve for J. We now know that ** J = 2 ($/lb) **

Because we've already solved for A in terms of J previously with Equation 3, we can plug in our value for J into this equation to solve for A.

Equation 3: A = (18/8) - (3/8)*J, plugging in J=2, we get that:

A = (18/8) - (3/8)*2, or A = (18/8) - (6/8). Because we already have a common denominator, we can combine these terms accordingly to get:

A = (12/8), or A = 1.5 ($/lb).

To summarize, we've now solved the system of equations for J=2 and A=1.5 In the context of our original problem, this means that each pound of jelly beans cost $2, and each pound of almonds cost $1.50. To check that our answers are correct, you can plug both A & J into either of the two original equations (Equation 1 and Equation 2). Because the answers on both sides of the equation agree (i.e. $18=$18, or $13=$13) we know that we've solved our systems of equations correctly.

Answer 2

3J + 8A = 18

5J + 2A = 13

multiply the 2nd by 4 to get

20J + 8A = 52, now subtract the 1st equation to get

17J = 34

J = 34/17 = $2 for jelly beans

3(2) +8A = 18

8A = 18-6 = 12

A = 12/8 = $1.50 for almonds