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 the value of x back into either equation to solve for y : y = 4 ( 5 ) − 17 .
Solve for y : y = 3 . The solution is ( 5 , 3 ) .
Explanation
Analyze 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 simultaneously. The equations are:
Equation 1: y = 6 x − 27 Equation 2: y = 4 x − 17
We will use the substitution method to solve this system. Since both equations are already solved for y , we can set them equal to each other.
Set equations equal Set the two equations equal to each other:
6 x − 27 = 4 x − 17
Solve for x Now, we solve for x . First, subtract 4 x from both sides of the equation:
6 x − 4 x − 27 = 4 x − 4 x − 17
2 x − 27 = − 17
Next, add 27 to both sides of the equation:
2 x − 27 + 27 = − 17 + 27
2 x = 10
Finally, divide both sides by 2:
2 2 x = 2 10
x = 5
Solve 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 2:
y = 4 x − 17
Substitute x = 5 :
y = 4 ( 5 ) − 17
y = 20 − 17
y = 3
Find the solution So, the solution to the system of equations is ( x , y ) = ( 5 , 3 ) .
Now we compare our solution to the multiple-choice options: a. ( − 5 , 3 ) b. ( − 3 , − 5 ) c. ( 5 , 3 ) d. No solution
The correct answer is (5, 3), which corresponds to option C.
Final Answer The solution to the system of equations is ( 5 , 3 ) . Therefore, the correct answer is C.
Examples
Systems of equations are used in various real-world applications. For example, they can be used to model supply and demand in economics, where the intersection of the supply and demand curves represents the equilibrium price and quantity. In engineering, systems of equations can be used to analyze circuits or structural systems. In everyday life, you might use a system of equations to determine the break-even point for a business venture, where costs equal revenue. Understanding how to solve systems of equations is a valuable skill in many fields.
The solution to the system of equations is ( 5 , 3 ) . Therefore, the correct answer is option c. By substituting the values, we verified the solution fulfills both equations.
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