Solve the system of equations:

[tex]
\begin{array}{l}
y=5 x+32 \
y=-4 x-22
\end{array}
[/tex]

A. (6,2)
B. (-6,-2)
C. (-6,2)
D. No solution

Set the two equations equal to each other: 5 x + 32 = − 4 x − 22 .
Solve for x : 9 x = − 54 , so x = − 6 .
Substitute x = − 6 into one of the equations to solve for y : y = 5 ( − 6 ) + 32 = 2 .
The solution to the system of equations is ( − 6 , 2 ) ​ .

Explanation

Analyze the problem We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously. The equations are:

Equation 1: y = 5 x + 32 Equation 2: y = − 4 x − 22

Set the equations equal Since both equations are solved for y , we can set them equal to each other to solve for x :

5 x + 32 = − 4 x − 22

Solve for x Now, we solve for x by adding 4 x to both sides and subtracting 32 from both sides:

5 x + 4 x = − 22 − 32
9 x = − 54
Divide both sides by 9 :
x = 9 − 54 ​
x = − 6

Solve for y Now that we have the value of x , we can substitute it back into either Equation 1 or Equation 2 to solve for y . Let's use Equation 1:

y = 5 x + 32
y = 5 ( − 6 ) + 32
y = − 30 + 32
y = 2

State the solution So, the solution to the system of equations is x = − 6 and y = 2 . We can express this as an ordered pair ( x , y ) = ( − 6 , 2 ) .

Select the correct answer Now, we compare our solution to the multiple-choice options:


a. ( 6 , 2 ) b. ( − 6 , − 2 ) c. ( − 6 , 2 ) d. No solution
Our solution ( − 6 , 2 ) matches option c.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company has a cost function C ( x ) = 20 x + 500 and a revenue function R ( x ) = 30 x , solving the system of equations y = 20 x + 500 and y = 30 x will give the number of units x that need to be sold for the company to break even, and the corresponding revenue y at that point. This helps businesses make informed decisions about pricing and production levels.

Answered by GinnyAnswer | 2025-07-03

By equating the two equations and solving, we find that the solution to the system is (-6, 2), which corresponds to option C.
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Answered by Anonymous | 2025-07-04