Solve the system by substitution. Check your solution.
$0.5 x+0.25 y=36$
$y+18=16 x$
a. $(36,72)$
c. $(49,81)$
b. $(9,126)$
d. $(21,9)$
Answer (1)
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 y = 16 x − 18 to find y : y = 126 . The solution is ( 9 , 126 ) .
Explanation
Analyze the problem 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 given options. The equations are:
0.5 x + 0.25 y = 36 y + 18 = 16 x
Solve for y First, we solve the second equation for y :
y = 16 x − 18
Substitute into the first equation Next, we substitute this expression for y into the first equation:
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 y = 16 x − 18 to find y :
y = 16 ( 9 ) − 18 y = 144 − 18 y = 126
Check the solution So the solution is ( x , y ) = ( 9 , 126 ) .
Now we check if this solution matches any of the given options. We see that it matches option b.
Final Answer Therefore, the solution to the system of equations is ( 9 , 126 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling traffic flow in a city. In this case, imagine you're managing a small business selling two products. The equations could represent constraints on resources or costs, and solving the system helps you find the production levels of each product that maximize profit or minimize expenses. Understanding how to solve these systems allows for better decision-making and resource allocation in many practical scenarios.
