Question

prerequisite: study the example showing how to solve problems with equations. then solve problems 1–8. example mrs. scott brings her car to a service station for repair. her bill shows a $215 charge for parts and a charge for 3\LXI0 hours of labor. the total bill is $495. write an equation to find the charge x for one hour of labor. charge for labor + charge for parts = total bill 3\\(\frac{1}{2}\\)x + 215 = 495 1 why is 3\\(\frac{1}{2}\\) multiplied by x in the equation? 2 how can you get 3\\(\frac{1}{2}\\)x alone on the left side of the equation 3\\(\frac{1}{2}\\)x + 215 = 495? what is the resulting equation? 3 multiply both sides of the simplified equation you wrote in problem 2 by a fraction to get x alone on the left side of the equation. simplify. 4 what is the charge for an hour of labor? 5 explain how to solve the equation \\(\frac{3}{4}\\)x + 5 = 11.

Explanation:

Problem 1

$x$ is the charge per hour of labor, and $3\frac{1}{2}$ is the number of labor hours. To find the total labor charge, we multiply the hourly rate ($x$) by the number of hours ($3\frac{1}{2}$), so $3\frac{1}{2}$ is multiplied by $x$ to represent the total labor cost.

Step1: Use subtraction property of equality

To get $3\frac{1}{2}x$ alone, we subtract 215 from both sides of the equation $3\frac{1}{2}x + 215 = 495$.
$$3\frac{1}{2}x + 215 - 215 = 495 - 215$$

Step2: Simplify both sides

Simplify the left - hand side: $3\frac{1}{2}x+(215 - 215)=3\frac{1}{2}x$.
Simplify the right - hand side: $495 - 215 = 280$.

Step1: Identify the coefficient of x

The coefficient of $x$ in the equation $3\frac{1}{2}x = 280$ is $3\frac{1}{2}=\frac{7}{2}$. To isolate $x$, we multiply both sides of the equation by the reciprocal of $\frac{7}{2}$, which is $\frac{2}{7}$.
$$\frac{2}{7}\times3\frac{1}{2}x=\frac{2}{7}\times280$$

Step2: Simplify the left - hand side

Since $\frac{2}{7}\times\frac{7}{2}x=x$ (because $\frac{2}{7}\times\frac{7}{2} = 1$).

Step3: Simplify the right - hand side

$\frac{2}{7}\times280=\frac{2\times280}{7}=2\times40 = 80$.

Answer:

Because $x$ is the charge for one hour of labor, and $3\frac{1}{2}$ is the number of hours of labor. Multiplying them gives the total charge for $3\frac{1}{2}$ hours of labor.

Problem 2