Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Ernest is a kickboxing instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of $\$99$ . Ernest will be earning a base salary of $\$91$ per month from the gym, plus an additional $\$4$ for every class he teaches. If Ernest teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take? How much will Ernest's expenses and earnings be? $\newline$ Once Ernest teaches _ classes, his expenses and earnings will both be $\$$ _.

View step-by-step help

Home

Math Problems

Precalculus

Solve a system of equations using substitution: word problems

Full solution

Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Ernest is a kickboxing instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of

\$99

. Ernest will be earning a base salary of

\$91

per month from the gym, plus an additional

\$4

for every class he teaches. If Ernest teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take? How much will Ernest's expenses and earnings be?

\newline

Once Ernest teaches _____ classes, his expenses and earnings will both be

\$

_____.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of classes Ernest teaches. $\newline$ Let $E$ be the total earnings after teaching $x$ classes. $\newline$ We know that Ernest's earnings ( $E$ ) are composed of a base salary ( $\$91$ ) plus $\$4$ for each class he teaches. $\newline$ So, we can write the equation for Ernest's earnings as: $\newline$ $E = 91 + 4x$
Equation for Earnings: We also know that Ernest wants to earn back the amount he spent on certification, which is $\$99$ . This means that his earnings ( $E$ ) should equal his expenses ( $\$99$ ) when he has taught enough classes. $\newline$ So, we can write the equation for Ernest's expenses as: $\newline$ $E = 99$
Equation for Expenses: Now we have a system of two equations: $\newline$ $1$ ) $E = 91 + 4x$ $\newline$ $2$ ) $E = 99$ $\newline$ We can solve this system using substitution. Since both equations equal $E$ , we can set them equal to each other: $\newline$ $91 + 4x = 99$
Solve for x: Next, we solve for $x$ by subtracting $91$ from both sides of the equation: $\newline$ $91 + 4x - 91 = 99 - 91$ $\newline$ $4x = 8$
Calculate $x$ : Now, we divide both sides by $4$ to find the value of $x$ : $\newline$ $\frac{4x}{4} = \frac{8}{4}$ $\newline$ $x = 2$
Total Earnings Calculation: Ernest needs to teach $2$ classes to earn back the amount he spent on certification. Now we can calculate his total earnings after teaching $2$ classes: $\newline$ $E = 91 + 4(2)$ $\newline$ $E = 91 + 8$ $\newline$ $E = 99$