Two factory plants are making TV panels. Yesterday, Plant A produced 3000 fewer panels than Plant B did. Three percent of the panels from Plant A and 5% of the panels from Plant B were defective. How many panels did Plant A produce, if the two plants together produced 1270 defective panels?

Question

theeric · Answer

Hi! It looks like you'll have to solve this like a "system of equations."

Pretty much, you have panels from Plant A and panels from Plant B, and you want to know how many panels came from each.

So the technique is $\rightarrow$ get all of your information into math form, and use variables for what you don't know. You don't know the numbers of panels. I'll say..
$\rightarrow$ You have $a$ Plant A panels
$\rightarrow$ You have $b$ Plant B panels.
Seems reasonable to me!

theeric · Answer

You have a ton of information. We can go sentence by sentence, though. Keep in mind that $a$ is the number of Plant A panels and $b$ is the number of Plant B panels.

theeric · Answer

"Yesterday, Plant A produced 3000 fewer panels than Plant B did."
So, You have $a$ panels, that were $3,000$ panels less than $b$ panels.
"Were" means they "were" equal.
$a=b-3,000$

theeric · Answer

"Three percent of the panels from Plant A ... were defective." So, 3% of $a$ was the number of defective panels. Since $3\%=.03$, there were $.03\ a$ defective panels from Plant A

theeric · Answer

"... 5% of the panels from Plant B were defective." Similarly to before, $5\%=.05$ so $.05\ b$ is the number of defective panels from Plant B.

theeric · Answer

"... the two plants together produced 1270 defective panels." 

We can start putting things together now.

So, the defective panels from Plant A plus the defective panels from Plant B equals $1,270$. Well, we have values for $\text{defective panels from Plant A}$ and $\text{defective panels from Plant B}$. They are $.03\ a$ and $.05\ b$ respectively, and they add up to $1,270$. Put into math speak, that's $.03\ a+.05\ b=1,270$

And we know from before that $a=b−3,000 $.


Now you have a couple equations.$$.03\ a+.05\ b=1,270$$$$a=b−3,000 $$

theeric · Answer

Now you can solve for $a$ in one equation, and substitute that into the other equation.

After you do that, you can solve for $b$. It will be the only variable. So then you'll know what $b$ is!

And, if you know what $b$ is, you can substitute its value in to the equation in which you solved for $a$. By doing that, you'll find the value of $a$.

That's the method for solving systems of equations. Solve for things, use substitution wisely, and keep on solving, calculating, and substituting until you find all your variables.