Choose the graph that matches the following system of equations:
$\begin{array}{l}
8 x+5 y=-5 \\
7 x-3 y=3
\end{array}$
Answer (1)
Solve the system of equations 8 x + 5 y = − 5 and 7 x − 3 y = 3 by eliminating y .
Multiply the first equation by 3 and the second by 5, then add them to get 59 x = 0 , so x = 0 .
Substitute x = 0 into the first equation to find y = − 1 .
The lines intersect at ( 0 , − 1 ) , so the correct graph shows this intersection. ( 0 , − 1 )
Explanation
Analyze the problem We are given a system of two linear equations:
8 x + 5 y = − 5
7 x − 3 y = 3
Our goal is to find the graph that corresponds to this system of equations. To do this, we can solve the system of equations to find the point of intersection.
Eliminate y and solve for x We can solve this system of equations using several methods, such as substitution or elimination. Let's use the elimination method.
Multiply the first equation by 3 and the second equation by 5 to eliminate y :
3 ( 8 x + 5 y ) = 3 ( − 5 ) ⇒ 24 x + 15 y = − 15
5 ( 7 x − 3 y ) = 5 ( 3 ) ⇒ 35 x − 15 y = 15
Now, add the two equations:
( 24 x + 15 y ) + ( 35 x − 15 y ) = − 15 + 15
59 x = 0
x = 0
Solve for y Now that we have the value of x , we can substitute it into either of the original equations to find the value of y . Let's use the first equation:
8 x + 5 y = − 5
8 ( 0 ) + 5 y = − 5
5 y = − 5
y = − 1
So, the point of intersection is ( 0 , − 1 ) .
Find the graph The solution to the system of equations is x = 0 and y = − 1 . This means that the two lines intersect at the point ( 0 , − 1 ) . The graph that matches this system of equations must have two lines that intersect at the point ( 0 , − 1 ) .
Final Answer The graph that matches the given system of equations is the one where the two lines intersect at the point (0, -1).
Conclusion Therefore, the graph that matches the system of equations is the one where the lines intersect at ( 0 , − 1 ) .
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 fixed costs of $10,000 and variable costs of $5 per unit, and they sell each unit for $15, the break-even point can be found by solving the system of equations:
Total Costs: C = 10000 + 5 x
Total Revenue: R = 15 x
Setting C = R gives 10000 + 5 x = 15 x , which simplifies to 10 x = 10000 , so x = 1000 units. This means the company needs to sell 1000 units to break even. Graphing these equations helps visualize the point where costs equal revenue.
