Solve the system of equations.
[tex]
\begin{array}{l}
y=5 x-8 \\
y=4 x-7
\end{array}
[/tex]
Answer (1)
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 to the system of equations is ( 1 , − 3 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
Equation 1: y = 5 x − 8
Equation 2: y = 4 x − 7
Our goal is to find the values of x and y that satisfy both equations simultaneously.
Equate the expressions for y Since both equations are already solved for y , we can set the expressions for y equal to each other. This gives us:
5 x − 8 = 4 x − 7
Solve 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
Solve 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 the first equation:
y = 5 x − 8
Substitute x = 1 :
y = 5 ( 1 ) − 8
y = 5 − 8
y = − 3
State the solution So, the solution to the system of equations is x = 1 and y = − 3 . This corresponds to the ordered pair ( 1 , − 3 ) .
Final Answer Therefore, 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 $8000 and variable costs of 5 p er u ni t , t h e t o t a l cos t c anb ere p rese n t e d a s y = 5x + 8000 , w h ere x$ is the number of units produced. If the company sells each unit for 9 , t h ere v e n u ec anb ere p rese n t e d a s y = 9x$. Solving this system of equations will give the number of units the company needs to sell to break even.
