A local hamburger shop sold a combined total of 558 hamburgers and cheeseburgers on Tuesday. There were 58 more cheeseburgers sold than hamburgers. How many hamburgers were sold on Tuesday?

Question

mathstudent55 · Answer

First, choose a variable for the number of hamburgers and another variable for the number of cheeseburgers.

mathstudent55 · Answer

Let h = number of hamburgers
Let c = number of cheese burgers

Ok so far?

lyssburleson · Answer

okay

mathstudent55 · Answer

You are told a total of 558 hamburgers and cheeseburgers combined were sold.
How do you represent the total number of hamburgers and cheeseburgers sold using h and c?

lyssburleson · Answer

wouldnt it be 
h+c=558?

mathstudent55 · Answer

You're a step ahead of me, but you're correct.
I was just looking for h + c, but you already correctly made that sum equal to 558.
You have an equation that shows the total number of hamburgers and cheeseburgers sold equal 558.

mathstudent55 · Answer

We have an equation with two unknowns.
We cannot solve for a unique solution.
There is an infinite number of solutions to this single equation with 2 unknowns.
For example, 500 and 58 are solutions, and so are 400 and 158, since each pair adds to 558.

We need a second equation.
We use the rest of the given info to come up with a second equation.

mathstudent55 · Answer

"There were 58 more cheeseburgers sold than hamburgers."
Can you use the variables h and c again to express the quote above mathematically?
Keep in mind that if 58 more cheeseburgers than hamburgers were sold, then the number of cheeseburgers is greater than the number of hamburgers by 58.

mathstudent55 · Answer

Example:
In a class there are 5 more girls than boys.
Number of boys = b, and number of girls = g.
Then g must be the larger number, so
g = b + 5
When you add 5 to b, the sum is greater than b, and we know g must be grater than b.

mathstudent55 · Answer

Now compare that situation with your problem.
c = cheeseburgers
h = hamburgers
There are 58 more cheeseburgers than hamburgers.
That means c must be grater than h.
If you add a positive number to h, you will get a lager number.
The positive number is 58 since there are 58 more cheeseburgers than hamburgers.

mathstudent55 · Answer

What do you add to h to find c, the number of cheeseburgers?

lyssburleson · Answer

so 58h+c=558?

mathstudent55 · Answer

No. Forget 558 for now.

All we are dealing with the second equation is:
The number of cheeseburgers is c.
The number of hamburgers is h
There are 58 more cheeseburgers than hamburgers.

lyssburleson · Answer

oh my bad. so h+58c

mathstudent55 · Answer

Examples:
If there are 5 hamburgers, then there are 5 + 58 cheeseburgers.
If there are 100 hamburgers, then there are 100 + 58 cheeseburgers.
If there are 300 hamburgers, then there are 300 + 58 cheeseburgers.

If there are h hamburgers, then there are _________ cheeseburgers.

What goes in the blank in the last line above?

mathstudent55 · Answer

You are close.
The answer is the number of cheeseburgers, c, is 58 more than the number of hamburgers, h, so the equation is

c = h + 58

mathstudent55 · Answer

Above you'd fill in the blank like this:

If there are h hamburgers, then there are __ h + 58 __ cheeseburgers.

mathstudent55 · Answer

Now we have two equations:

c + h = 558
c = h + 58

mathstudent55 · Answer

Now we have a system of two equations and two unknowns.
Now we can solve the system of equations to find a unique solution.

mathstudent55 · Answer

Are you familiar with different methods of solving systems of equations?

mathstudent55 · Answer

I have to go, so I'll just explain one method.
If you have questions, just post them, and I;'ll try to answer them when I get back.

mathstudent55 · Answer

One method is the Substitution Method.
In this method, you solve one equation for one variable, then you substitute what that variable is equal to in the other equation.
I picked this method for this problem because the second equation we got is already solved for c.

mathstudent55 · Answer

c + h = 558
c = h + 58

Since c = h + 58, we replace c of the first equation with h + 58

$\color{red}{c} + h = 558$
$c = \color{red}{h + 58}$

Now we rewrite the first equation, replacing c with what c is equal to from the second equation:

$\color{red}{h + 58} + h = 558$

Add like terms:

$2h + 58 = 558$

Subtract 58 from both sides:

$2h = 500$

Divide both sides by 2:

$h = 250$

The answer is:

250 hamburgers were sold on Tuesday.