# Word Problems – System Of Equations

To solve a word problem using a system of equations, it is important to;
– Identify what we don’t know
– Declare variables
– Use sentences to create equations

An example on how to do this:

Mary and Jose each bought plants from the same store. Mary spent \$188 on 7 cherry trees and 11 rose bushes. Jose spent \$236 on 13 cherry trees and 11 rose bushes. Find the cost of one cherry tree and the cost of one rose bush.

Cost of a cherry tree: $x$
Cost of a rose bush: $y$
7 cherry trees and 11 rose bushes = \$188 $7x + 11y = 188$ $13x + 11y = 236$ $7x + 11y = 188$ $\underline {-13x - 11y = -236 }$

The y-values cancel each other out, so now you are left with only x-values and real numbers. $-6x = -48$ $x = 8$

Then, you plug in your x-value into an original equation in order to find the y-value. $7x + 11y = 188$ $7(8) + 11y = 188$ $56 + 11y = 188$ $11y = 132$ $y = 12$

Cost of a cherry tree: \$8
Cost of a rose bush: \$12

## Video-Lesson Transcript

To solve a word problem using system of equations, it is important to:
1. Identify what we don’t know
2. Declare variables.
3. Use sentences to create equations.

Let’s have an example:

Mary and Jose each bought plants from the same store. Mary spent $\188$ on $7$ cherry trees and $11$ rose bushes. Jose spent $\236$ on $13$ cherry trees and $11$ rose bushes. Find the cost of one cherry tree and the cost of one rose bush.

Let’s solve this by following steps above.

1. What we don’t know:
cost of a cherry tree
cost of a rose bush

2. Declare variables:
cost of a cherry tree : $x$
cost of a rose bush : $y$

3. Use sentences to create equations.

For Mary: $7$ cherry trees and $11$ rose bushes $= \188$ $7x + 11 y = 188$

For Jose: $13$ cherry trees and $11$ rose bushes $= \236$ $13x + 11 y = 236$

Now, we have a system of equations $7x + 11y = 188$ $13x + 11y = 236$

We can solve this by process of substitution, elimination or fraction.

Since the value of $y$ is the same for both equations, let’s do the process of elimination.

First, let’s multiply the first equation by $-1$ $-1 (13x + 11y = 236)$

Here we’ll have a negated equation $-13x - 11y = -236$

Let’s do the process of elimination now $7x + 11y = 188$ $+ {-13x} + 11y = 236$

We’ll have $6x = -48$

Then, let’s isolate $x$ by dividing both sides by $-6$ $\dfrac{-6x}{-6} = \dfrac{-48}{-6}$

Now, we have $x = 8$

Remember, our declared variable?

cost of a cherry tree : $x$

Since $x = 8$

Now we can say that

cost of a cherry tree : $= 8$ Now, let’s solve for the value of $y$ by getting one equation and plugging the value of $x$.

Let’s use the first equation to plug in $x = 8$ $7x + 11y = 188$ $7(8) + 11y = 188$ $56 + 11y = 188$

Let’s isolate $y$ by subtracting $56$ on both sides of the equation $56 - 56 + 11y = 188 - 56$ $11y = 132$

Then divide by $11$ $\dfrac{11y}{11} = \dfrac{132}{11}$

And we get $y = 12$

Now, we know that cost of a rose bush is $12$.