Solve the system by substitution. Check your solution.
[tex]0.5 x+0.25 y=36[/tex]
[tex]y+18=16 x[/tex]
a. [tex](36,72)[/tex]
c. [tex](49,81)[/tex]
b. [tex](9,126)[/tex]
d. [tex](21,9)[/tex]
Please select the best answer from the choices provided
Answer (2)
Solve the second equation for y : y = 16 x − 18 .
Substitute the expression for y into the first equation: 0.5 x + 0.25 ( 16 x − 18 ) = 36 .
Simplify and solve for x : x = 9 .
Substitute the value of x back into the equation for y : y = 126 . The solution is ( 9 , 126 ) .
Explanation
Problem Analysis We are given a system of two equations with two variables, x and y . Our goal is to solve this system using the substitution method and identify the correct solution from the provided options.
Given Equations The given equations are:
Equation 1: 0.5 x + 0.25 y = 36
Equation 2: y + 18 = 16 x
Solve for y First, we solve Equation 2 for y :
y = 16 x − 18
Substitution Next, we substitute this expression for y into Equation 1:
0.5 x + 0.25 ( 16 x − 18 ) = 36
Solve for x Now, we simplify and solve for x :
0.5 x + 4 x − 4.5 = 36
4.5 x = 40.5
x = 4.5 40.5 = 9
Solve for y Substitute the value of x back into the equation for y :
y = 16 ( 9 ) − 18 = 144 − 18 = 126
Check the Solution So, the solution is ( x , y ) = ( 9 , 126 ) . Now, let's check this solution in both equations to make sure it is correct.
Equation 1: 0.5 ( 9 ) + 0.25 ( 126 ) = 4.5 + 31.5 = 36 (Correct)
Equation 2: 126 + 18 = 144 = 16 ( 9 ) (Correct)
Final Answer The solution to the system of equations is ( 9 , 126 ) . Comparing this to the given options, we see that it matches option b.
Conclusion Therefore, the correct answer is b. ( 9 , 126 )
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, a company might use a system of equations to model its costs and revenues, and then solve the system to find the production level at which costs equal revenues. This helps the company make informed decisions about pricing, production, and resource allocation. Understanding how to solve systems of equations is a valuable skill in many fields.
The solution to the system of equations is (9, 126), found by substituting and solving. Both equations are satisfied with this solution. Thus, the correct choice from the options is b. (9, 126).
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