Consider the system of equations.
[tex]y=-2 x+4[/tex]
[tex]y=-\frac{1}{3} x-1[/tex]
The solution is [tex](3,-2)[/tex].
Verify the solution. Which true statement appears in your check?
A. [tex]4=4[/tex]
B. [tex]-2=-2[/tex]
C. [tex]-1=-1[/tex]
D. [tex]-3=-3[/tex]
Answer (2)
The solution (3, -2) satisfies both equations in the system. The true statement that appears in our verification is -2 = -2. Hence, the correct choice is B.
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Substitute the solution ( 3 , − 2 ) into the first equation: y = − 2 x + 4 , which yields − 2 = − 2 ( 3 ) + 4 .
Simplify the first equation: − 2 = − 6 + 4 , resulting in − 2 = − 2 .
Substitute the solution ( 3 , − 2 ) into the second equation: y = − 3 1 x − 1 , which yields − 2 = − 3 1 ( 3 ) − 1 .
Simplify the second equation: − 2 = − 1 − 1 , resulting in − 2 = − 2 . The true statement is − 2 = − 2 .
Explanation
Problem Analysis We are given a system of equations and a proposed solution ( 3 , − 2 ) . Our task is to verify whether this solution satisfies both equations and to identify which true statement appears during the verification process.
Substituting into the First Equation First, we substitute x = 3 and y = − 2 into the first equation: y = − 2 x + 4
− 2 = − 2 ( 3 ) + 4
Simplifying the First Equation Simplifying the right side of the equation, we get: − 2 = − 6 + 4
− 2 = − 2
This is a true statement.
Substituting into the Second Equation Next, we substitute x = 3 and y = − 2 into the second equation: y = − 3 1 x − 1
− 2 = − 3 1 ( 3 ) − 1
Simplifying the Second Equation Simplifying the right side of the equation, we get: − 2 = − 1 − 1
− 2 = − 2
This is also a true statement.
Conclusion Both equations are satisfied by the solution ( 3 , − 2 ) . The true statement that appears in our check is − 2 = − 2 .
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's cost function is y = 5 x + 100 (where x is the number of units produced and y is the total cost) and the revenue function is y = 15 x , solving this system of equations will give the number of units the company needs to sell to break even. In this case, solving the system gives x = 10 and y = 150 , meaning the company breaks even when it sells 10 units and has a total cost and revenue of $150.
