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]
B. [tex](9,126)[/tex]
C. [tex](49,81)[/tex]
D. [tex](21,9)[/tex]
Answer (2)
Solve the second equation for y : y = 16 x − 18 .
Substitute the expression for y into the first equation 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 to the system of equations is ( 9 , 126 ) , which corresponds to option b. ( 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 then choose the correct solution from the provided options.
The given equations are:
0.5 x + 0.25 y = 36
y + 18 = 16 x
We will solve for y in the second equation and substitute that expression into the first equation.
Solve for y in the second equation First, let's solve the second equation for y :
y + 18 = 16 x
Subtract 18 from both sides:
y = 16 x − 18
Substitute and solve for x Now, substitute the expression for y into the first equation:
0.5 x + 0.25 ( 16 x − 18 ) = 36
Distribute the 0.25 :
0.5 x + 4 x − 4.5 = 36
Combine like terms:
4.5 x − 4.5 = 36
Add 4.5 to both sides:
4.5 x = 40.5
Divide by 4.5 :
x = 4.5 40.5 = 9
Solve for y Now that we have the value of x , we can substitute it back into the equation y = 16 x − 18 to find the value of y :
y = 16 ( 9 ) − 18
y = 144 − 18
y = 126
Check the solution and choose the correct option So, the solution to the system of equations is ( x , y ) = ( 9 , 126 ) .
Now, let's check which of the given options matches our solution:
a. ( 36 , 72 ) b. ( 9 , 126 ) c. ( 49 , 81 ) d. ( 21 , 9 )
The correct solution is option b. ( 9 , 126 ) .
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell their 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. Similarly, in physics, systems of equations can be used to analyze the forces acting on an object in equilibrium. In chemistry, they can be used to balance chemical equations.
The solution to the system of equations is ( 9 , 126 ) . This was verified by substituting the values of x and y back into the original equations, where both hold true. Therefore, the correct choice is option B.
;
