Use the elimination method to solve the system of equations. Choose the correct ordered pair.
[tex]\begin{array}{l}
7 x+3 y=30 \\
-2 x+3 y=3
\end{array}[/tex]
A. (3,3)
B. (3,5)
C. (6,5)
D. (6,3)
Answer (1)
Subtract the second equation from the first to eliminate y : 9 x = 27 .
Solve for x : x = 3 .
Substitute x = 3 into the first equation: 7 ( 3 ) + 3 y = 30 .
Solve for y : y = 3 . The solution is ( 3 , 3 ) .
Explanation
Analyze the problem We are given the following system of equations:
7 x + 3 y = 30 − 2 x + 3 y = 3
Our goal is to solve for x and y using the elimination method.
Eliminate y To eliminate y , we can subtract the second equation from the first equation:
( 7 x + 3 y ) − ( − 2 x + 3 y ) = 30 − 3
Simplifying this, we get:
7 x + 3 y + 2 x − 3 y = 27
9 x = 27
Solve for x Now, we solve for x :
x = 9 27 = 3
Substitute x into the first equation Next, we substitute the value of x into one of the original equations to solve for y . Let's use the first equation:
7 x + 3 y = 30
7 ( 3 ) + 3 y = 30
21 + 3 y = 30
Solve for y Now, we solve for y :
3 y = 30 − 21
3 y = 9
y = 3 9 = 3
Find the correct ordered pair The solution to the system of equations is the ordered pair ( 3 , 3 ) .
Comparing this to the given options, we see that the correct answer is A. ( 3 , 3 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs per unit, and they sell each unit at a certain price, we can set up 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.
