Solve the system of equations:
[tex]
\begin{array}{l}
y=6 x-27 \\
y=4 x-17
\end{array}
[/tex]
A. (-5,3)
B. (-3,-5)
C. (5,3)
D. No solution
Answer (2)
Set the two equations equal to each other: 6 x − 27 = 4 x − 17 .
Solve for x : 2 x = 10 , so x = 5 .
Substitute x = 5 into one of the equations to solve for y : y = 4 ( 5 ) − 17 = 3 .
The solution to the system of equations is ( 5 , 3 ) , so the answer is ( 5 , 3 ) .
Explanation
Understanding the Problem We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations. The equations are:
Equation 1: y = 6 x − 27 Equation 2: y = 4 x − 17
Setting the Equations Equal Since both equations are solved for y , we can set them equal to each other:
6 x − 27 = 4 x − 17
Isolating x Now, we solve for x . First, subtract 4 x from both sides:
6 x − 4 x − 27 = 4 x − 4 x − 17
2 x − 27 = − 17
Continuing to Isolate x Next, add 27 to both sides:
2 x − 27 + 27 = − 17 + 27
2 x = 10
Solving for x Finally, divide both sides by 2:
2 2 x = 2 10
x = 5
Solving for y Now that we have the value of x , we can substitute it into either equation to find the value of y . Let's use Equation 2:
y = 4 x − 17
y = 4 ( 5 ) − 17
y = 20 − 17
y = 3
Stating the Solution So the solution to the system of equations is x = 5 and y = 3 . Therefore, the solution is ( 5 , 3 ) .
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company's cost function is C ( x ) = 4 x + 17 and its revenue function is R ( x ) = 6 x + 27 , solving the system of equations allows the company to find the number of units x for which the cost equals the revenue, indicating the break-even point. Understanding how to solve systems of equations is crucial for making informed business decisions and optimizing resource allocation.
The solution to the system of equations y = 6 x − 27 and y = 4 x − 17 is obtained by setting them equal to each other, solving for x , and then substituting back to find y . The final solution is ( 5 , 3 ) , so the chosen option is C. (5, 3) .
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