Solve the system of equations:
[tex]
\begin{array}{l}
y=5 x-8 \\
y=4 x-7
\end{array}
[/tex]
A. (-1,-3)
B. (-1,3)
C. (1,-3)
D. No solution
Answer (2)
Set the two equations equal to each other: 5 x − 8 = 4 x − 7 .
Solve for x : x = 1 .
Substitute x = 1 into one of the equations to solve for y : y = 5 ( 1 ) − 8 = − 3 .
The solution is ( 1 , − 3 ) , so the answer is ( 1 , − 3 ) .
Explanation
Setting up the Equations We are given a system of two linear equations:
y = 5 x − 8 y = 4 x − 7
Our goal is to find the values of x and y that satisfy both equations simultaneously. We can solve this system by setting the two expressions for y equal to each other.
Equating the Expressions Since both equations are solved for y , we can set the right-hand sides equal to each other:
5 x − 8 = 4 x − 7
Solving for x Now, we solve for x . Subtract 4 x from both sides:
5 x − 4 x − 8 = 4 x − 4 x − 7 x − 8 = − 7
Add 8 to both sides:
x − 8 + 8 = − 7 + 8 x = 1
Solving for y Now that we have the value of x , we can substitute it back into either of the original equations to find the value of y . Let's use the first equation:
y = 5 x − 8 y = 5 ( 1 ) − 8 y = 5 − 8 y = − 3
Finding the Correct Option So, the solution to the system of equations is x = 1 and y = − 3 . This corresponds to the point ( 1 , − 3 ) . Now we compare our solution to the given choices:
a. ( − 1 , − 3 ) b. ( − 1 , 3 ) c. ( 1 , − 3 ) d. No solution
The correct answer is c. ( 1 , − 3 ) .
Examples
Systems of equations are used in various real-world applications. For example, when planning a budget, you might have one equation representing your income and another representing your expenses. Solving the system helps you determine the break-even point. Similarly, in physics, systems of equations can describe the motion of objects, and solving them allows you to predict their future positions. In business, these systems can help optimize production and pricing strategies to maximize profit.
The solution to the system of equations is found by setting the two equations equal, leading to x = 1 and substituting this value gives y = − 3 . Thus, the solution is the point ( 1 , − 3 ) . The correct answer choice is C, ( 1 , − 3 ) .
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