I need someone to explain how to solve equations with two variables and tell me what the formula or equation is?

Please help and explain. Thank you!

what does the problem look like?

this is what the problem says

Find five pairs of numbers for shirt and cap sales that will allow the students to make a $600 profit exactly.

I want to know how to write the equation/formula and how to solve it

46889

To solve equations with two variables, we often use the method of substitution or elimination. Let me explain both methods to you.

Method 1: Substitution
1. Start by solving one of the equations for one variable in terms of the other variable.
For example, if you have the equations:
Equation 1: 2x + 3y = 7
Equation 2: 4x - y = 5

Solve Equation 2 for y:
4x - y = 5
-y = 5 - 4x
y = 4x - 5

2. Substitute the value of y from the first equation into the second equation.

Substitute y = 4x - 5 into Equation 1:
2x + 3(4x - 5) = 7

3. Simplify and solve for x:
2x + 12x - 15 = 7
14x = 22
x = 22/14
x = 11/7

4. Substitute the value of x back into either equation to find the value of y:
Using Equation 1:
2(11/7) + 3y = 7
22/7 + 3y = 7
3y = 49/7 - 22/7
3y = 27/7
y = 27/7 * 1/3
y = 9/7

So the solution to the system of equations is x = 11/7 and y = 9/7.

Method 2: Elimination
1. Multiply one or both of the equations by a constant to ensure that the coefficients of one of the variables in both equations are opposites.

Let's use the given equations again:
Equation 1: 2x + 3y = 7
Equation 2: 4x - y = 5

Multiply Equation 2 by 3 to make the coefficients of y opposites:
12x - 3y = 15

2. Add the two equations together to eliminate one variable:
(2x + 3y) + (12x - 3y) = 7 + 15
2x + 12x = 22

3. Simplify and solve for x:
14x = 22
x = 22/14
x = 11/7

4. Substitute the value of x back into either equation to find the value of y:
Using Equation 1:
2(11/7) + 3y = 7
22/7 + 3y = 7
3y = 49/7 - 22/7
3y = 27/7
y = 27/7 * 1/3
y = 9/7

Again, the solution to the system of equations is x = 11/7 and y = 9/7.

The formula or equation for solving a system of equations with two variables is not a single equation but rather a combination of algebraic methods to eliminate or substitute variables, as shown in the example above.