Use the substitution method to solve the system of equations. Choose the correct ordered pair.
[tex]$\begin{array}{l}
2 x+2 y=16 \\
y=x-4
\end{array}$[/tex]
A. (2,-2)
B. (6,-2)
C. (2,2)
D. (6,2)
Answer (1)
Substitute y = x − 4 into 2 x + 2 y = 16 .
Simplify and solve for x : 2 x + 2 ( x − 4 ) = 16 ⇒ 4 x − 8 = 16 ⇒ 4 x = 24 ⇒ x = 6 .
Substitute x = 6 into y = x − 4 to find y : y = 6 − 4 = 2 .
The solution is ( 6 , 2 ) .
Explanation
Analyze the problem We are given the following system of equations:
{ 2 x + 2 y = 16 y = x − 4
Our goal is to solve this system using the substitution method and identify the correct ordered pair from the given options.
Substitution Substitute the expression for y from the second equation into the first equation:
2 x + 2 ( x − 4 ) = 16
Now, we solve for x .
Solve for x Distribute the 2 in the equation:
2 x + 2 x − 8 = 16
Combine like terms:
4 x − 8 = 16
Add 8 to both sides:
4 x = 24
Divide by 4 :
x = 6
So, x = 6 .
Solve for y Now that we have the value of x , we can substitute it back into the second equation to find the value of y :
y = x − 4
y = 6 − 4
y = 2
So, y = 2 .
Final Answer The solution to the system of equations is the ordered pair ( x , y ) = ( 6 , 2 ) . Comparing this with the given options, we see that it matches option D.
Therefore, the correct ordered pair is ( 6 , 2 ) .
Examples
Systems of equations are used in various real-life situations, 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, they can use a system of equations to find the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve systems of equations is crucial for making informed business decisions.
