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 is ( 5 , 3 ) , which means the answer is ( 5 , 3 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = 6 x − 27
y = 4 x − 17
Our goal is to find the values of x and y that satisfy both equations simultaneously.
Setting the Equations Equal Since both equations are already solved for y , we can set the expressions for y equal to each other:
6 x − 27 = 4 x − 17
Solving for x Now, we solve for x . Subtract 4 x from both sides:
6 x − 4 x − 27 = 4 x − 4 x − 17 2 x − 27 = − 17
Add 27 to both sides:
2 x − 27 + 27 = − 17 + 27 2 x = 10
Divide by 2:
x = 2 10 x = 5
Solving 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 second equation:
y = 4 x − 17 y = 4 ( 5 ) − 17 y = 20 − 17 y = 3
The Solution So, the solution to the system of equations is x = 5 and y = 3 . We can write this as the ordered pair ( 5 , 3 ) .
Checking the Solution To check our solution, we substitute x = 5 and y = 3 into both equations:
Equation 1: y = 6 x − 27 3 = 6 ( 5 ) − 27 3 = 30 − 27 3 = 3 (True)
Equation 2: y = 4 x − 17 3 = 4 ( 5 ) − 17 3 = 20 − 17 3 = 3 (True)
Since the solution satisfies both equations, it is correct.
Final Answer The solution to the system of equations 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 has fixed costs and variable costs, and they sell a product at a certain price, a system of equations can be set up to find the number of units they need to sell to cover their costs. Another example is in physics, where systems of equations can be used to solve for unknown forces or velocities in a system.
The solution to the system of equations is ( 5 , 3 ) . We found this by setting the equations equal, solving for x , and then finding y . The chosen multiple choice option is C. (5, 3).
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