Solve the system of equations:
[tex]
\begin{array}{l}
y=-5 x-8 \\
y=4 x+1
\end{array}
[/tex]
A. (1, -3)
B. (-1, -3)
C. (3, 1)
D. No solution
Answer (2)
Set the two equations equal to each other: − 5 x − 8 = 4 x + 1 .
Solve for x : x = − 1 .
Substitute the value of x into one of the equations to solve for y : y = 4 ( − 1 ) + 1 = − 3 .
The solution to the system of equations is ( − 1 , − 3 ) .
Explanation
Setting up the equations We are given a system of two linear equations:
Equation 1: y = − 5 x − 8 Equation 2: y = 4 x + 1
We need to find the values of x and y that satisfy both equations. Since both equations are solved for y , we can set the expressions for y equal to each other.
Solving for x Setting the expressions for y equal to each other, we get: − 5 x − 8 = 4 x + 1 Now, we solve for x . Add 5 x to both sides: − 8 = 9 x + 1 Subtract 1 from both sides: − 9 = 9 x Divide both sides by 9: x = − 1
Solving for y Now that we have the value of x , we can substitute it back into either equation to solve for y . Let's use Equation 2: y = 4 x + 1 Substitute x = − 1 :
y = 4 ( − 1 ) + 1 y = − 4 + 1 y = − 3
Finding the solution So the solution to the system of equations is x = − 1 and y = − 3 . Therefore, the solution is ( − 1 , − 3 ) .
Comparing this to the given options, we see that it matches option b.
Final Answer The solution to the system of equations is ( − 1 , − 3 ) .
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, if a company has fixed costs of $8000 and variable costs of $5 per unit, and they sell each unit for 13 , w ec an se t u p a sys t e m o f e q u a t i o n s t o f in d t h e n u mb ero f u ni t s t h ey n ee d t ose llt o b re ak e v e n . L e t y b e t h e t o t a l cos t an d re v e n u e , an d x b e t h e n u mb ero f u ni t s . T h ecos t e q u a t i o ni s y = 5x + 8000 , an d t h ere v e n u ee q u a t i o ni s y = 13x$. Solving this system of equations will give the break-even point.
The solution to the system of equations is (-1, -3), which corresponds to option B. This was found by setting the equations equal to each other, solving for x, and then finding y. Verification shows both equations confirm this point.
;
