Mrs. Smith gave her Algebra class the following linear equation as part of their warm-up.
[tex]19-9 x=22[/tex]
Each student was given a different linear equation to solve and then compare their solution with the original equation. Maryanne and Herb were given the equations below.
[tex]
\begin{aligned}
\text { Maryanne: } & -\frac{12}{5} x+\frac{26}{5}=6 \\
\text { Herb: } & 11=18 x+5
\end{aligned}
[/tex]
Select the correct comparison of the three equations.
A. Both Maryanne's and Herb's equations have the same solution as the original.
B. Only Maryanne's equation has the same solution as the original equation.
C. Both Maryanne's and Herb's equations have a different solution than the original.
D. Only Herb's equation has the same solution as the original equation.
Answer (1)
Solve the original equation 19 − 9 x = 22 and find x = − 3 1 .
Solve Maryanne's equation − 5 12 x + 5 26 = 6 and find x = − 3 1 .
Solve Herb's equation 11 = 18 x + 5 and find x = 3 1 .
Only Maryanne's equation has the same solution as the original equation: B .
Explanation
Problem Analysis We are given three linear equations and asked to compare their solutions. The first equation is the original equation, and the other two are equations given to Maryanne and Herb. We need to solve each equation for x and then compare the solutions to determine which equations have the same solution as the original equation.
Solving the Original Equation First, let's solve the original equation: 19 − 9 x = 22 Subtract 19 from both sides: − 9 x = 22 − 19 − 9 x = 3 Divide both sides by -9: x = − 9 3 x = − 3 1
Solving Maryanne's Equation Next, let's solve Maryanne's equation: − 5 12 x + 5 26 = 6 Subtract 5 26 from both sides: − 5 12 x = 6 − 5 26 − 5 12 x = 5 30 − 5 26 − 5 12 x = 5 4 Multiply both sides by − 12 5 :
x = 5 4 × − 12 5 x = − 5 × 12 4 × 5 x = − 60 20 x = − 3 1
Solving Herb's Equation Now, let's solve Herb's equation: 11 = 18 x + 5 Subtract 5 from both sides: 11 − 5 = 18 x 6 = 18 x Divide both sides by 18: x = 18 6 x = 3 1
Comparing the Solutions Comparing the solutions, we have: Original equation: x = − 3 1 Maryanne's equation: x = − 3 1 Herb's equation: x = 3 1
We can see that the original equation and Maryanne's equation have the same solution, which is x = − 3 1 . Herb's equation has a different solution, which is x = 3 1 .
Conclusion Therefore, only Maryanne's equation has the same solution as the original equation.
Examples
When designing a bridge, engineers use linear equations to model the distribution of weight and stress. If the original design equation is 15 x + 20 = 50 , and two alternative designs are represented by − 2 5 x + 5 = 2 15 and 3 x − 2 = 4 , solving these equations helps determine if the alternative designs maintain the same structural integrity as the original. Ensuring the solutions match guarantees that the bridge will safely bear the intended load, preventing potential collapses or failures. This ensures public safety and efficient resource allocation.
