Solve the system of equations.
[tex]
\begin{array}{l}
y=7 x-53 \\
y=-4 x+35
\end{array}
[/tex]
a. (-3,-8)
b. (8,3)
c. (-8,3)
d. No solution
Answer (1)
Set the two equations equal to each other: 7 x − 53 = − 4 x + 35 .
Solve for x : 11 x = 88 , so x = 8 .
Substitute x = 8 into either equation to solve for y : y = 7 ( 8 ) − 53 = 3 .
The solution to the system of equations is ( 8 , 3 ) .
Explanation
Problem Analysis We are given a system of two linear equations:
Equation 1: y = 7 x − 53 Equation 2: y = − 4 x + 35
Our goal is to find the values of x and y that satisfy both equations simultaneously.
Setting Equations Equal Since both equations are already solved for y , we can set the expressions for y equal to each other:
7 x − 53 = − 4 x + 35
Solving for x Now, we solve for x . Add 4 x to both sides of the equation:
7 x + 4 x − 53 = − 4 x + 4 x + 35 11 x − 53 = 35
Add 53 to both sides:
11 x − 53 + 53 = 35 + 53 11 x = 88
Divide both sides by 11 :
x = 11 88 x = 8
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 1:
y = 7 x − 53 y = 7 ( 8 ) − 53 y = 56 − 53 y = 3
Solution So, the solution to the system of equations is x = 8 and y = 3 . We can write this as the ordered pair ( 8 , 3 ) .
Verification To ensure our solution is correct, we can check it in both equations:
Equation 1: y = 7 x − 53 3 = 7 ( 8 ) − 53 3 = 56 − 53 3 = 3 (Correct)
Equation 2: y = − 4 x + 35 3 = − 4 ( 8 ) + 35 3 = − 32 + 35 3 = 3 (Correct)
Since the solution satisfies both equations, it is indeed the correct solution.
Final Answer Comparing our solution ( 8 , 3 ) with the given options, we find that it matches option b.
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 how much you can save or need to cut back. In business, systems of equations can help optimize production costs or determine the break-even point for a product. They also appear in physics, engineering, and computer science for modeling and solving complex problems.
