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
Answer (1)
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
Understanding the Problem We are given a system of two linear equations:
Equation 1: y = 5 x + 32 Equation 2: y = − 4 x − 22
Our goal is to find the values of x and y that satisfy both equations simultaneously.
Setting Equations Equal Since both equations are solved for y , we can set the expressions for y equal to each other:
5 x + 32 = − 4 x − 22
Solving for x Now, we solve for x . Add 4 x to both sides:
5 x + 4 x + 32 = − 4 x + 4 x − 22 9 x + 32 = − 22
Subtract 32 from both sides:
9 x + 32 − 32 = − 22 − 32 9 x = − 54
Divide by 9:
x = 9 − 54 x = − 6
Solving for y Now that we have the value of x , we can substitute it back into either equation to find the value of y . Let's use Equation 1:
y = 5 x + 32 y = 5 ( − 6 ) + 32 y = − 30 + 32 y = 2
Final Answer So the solution is x = − 6 and y = 2 . Therefore, the solution to the system of equations is ( − 6 , 2 ) .
Examples
Systems of equations are used in real life to model situations where there are multiple constraints or conditions. For example, imagine you are trying to determine the optimal pricing strategy for a product. You might have one equation that represents the relationship between price and demand, and another equation that represents the relationship between price and cost. Solving this system of equations would help you find the price that maximizes your profit, balancing demand and cost considerations. This is just one of many applications of systems of equations in economics, engineering, and other fields.
