Solve the system by substitution. Check your solution.
[tex]\begin{array}{l}
y=0.8 x+7.2 \
20 x+32 y=48\end{array}[/tex]
a. (8,-6)
b. (5,11)
c. (-4,4)
d. (-10,2)
Answer (2)
Substitute the first equation into the second equation: 20 x + 32 ( 0.8 x + 7.2 ) = 48 .
Solve for x : 20 x + 25.6 x + 230.4 = 48 ⇒ 45.6 x = − 182.4 ⇒ x = − 4 .
Substitute x = − 4 into the first equation to find y : y = 0.8 ( − 4 ) + 7.2 = 4 .
The solution is ( − 4 , 4 ) , which matches option c. ( − 4 , 4 )
Explanation
Problem Analysis We are given the following system of equations:
y = 0.8 x + 7.2
20 x + 32 y = 48
Our goal is to solve this system using the substitution method and then check our solution against the provided options.
Substitution We will substitute the first equation into the second equation to eliminate y :
20 x + 32 ( 0.8 x + 7.2 ) = 48
Now, we solve for x .
Solving for x Expanding the equation, we get:
20 x + 25.6 x + 230.4 = 48
Combining like terms:
45.6 x = 48 − 230.4 45.6 x = − 182.4
Dividing both sides by 45.6:
x = 45.6 − 182.4 = − 4
Solving for y Now that we have the value of x , we can substitute it back into the first equation to find the value of y :
y = 0.8 ( − 4 ) + 7.2 y = − 3.2 + 7.2 y = 4
Solution So, the solution to the system of equations is x = − 4 and y = 4 . This corresponds to the point ( − 4 , 4 ) .
Checking the Solution Now, let's check our solution by substituting x = − 4 and y = 4 into both original equations:
Equation 1:
4 = 0.8 ( − 4 ) + 7.2 4 = − 3.2 + 7.2 4 = 4
Equation 2:
20 ( − 4 ) + 32 ( 4 ) = 48 − 80 + 128 = 48 48 = 48
Since the solution satisfies both equations, it is correct.
Final Answer Comparing our solution ( − 4 , 4 ) with the given options, we find that it matches option c.
Conclusion Therefore, the correct answer is c. ( − 4 , 4 ) .
Examples
Systems of equations are used in various real-life scenarios, 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. Another example is in electrical engineering, where systems of equations are used to analyze circuits and determine the currents and voltages at different points in the circuit. These applications demonstrate the practical importance of understanding and solving systems of equations.
The solution to the system of equations is (-4, 4), which matches option c. This was found by substituting the expression for y into the second equation, solving for x, and then substituting back to find y. The solution was verified by checking it against both original equations.
;
