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A theater group made appearances in two cities. The hotel charge before tax in the second city...

Question:

A theater group made appearances in two cities. The hotel charge before tax in the second city was $1500 higher than in the first. The tax in the first city was 8%, and the tax in the second city was 6%. The total hotel tax paid for the two cities was $650. How much was the hotel charge in each city before tax?

System of Equations:

The system of equations or equation system contains at least two the same set of variables. This set of equations are typically solved by the substitution method. By this method,

  • We substitute one equation in terms of a particular variable in another equation to solve for another unknown variable.
  • By using the result of the variable, we obtain the unknown value of a remaining variable.

Answer and Explanation: 1

Let {eq}x {/eq} be the hotel charge before tax in the first city, and {eq}y {/eq} be the hotel charge before tax in the second city.

Given statement:

  • The hotel charge before tax in the second city was {eq}\$ 1500 {/eq} higher than in the first.

$$y = x + 1500 \tag{Eq. 1} $$

  • The tax in the first city was {eq}8 \% {/eq}, and the tax in the second city was {eq}6 \% {/eq}. The total hotel tax paid for the two cities was {eq}\$ 650 {/eq}.

$$8 \% x + 6 \% y = 650 \tag{Eq. 2} $$

Substitute the {eq}\text{Eq. 1} {/eq} in the {eq}\text{Eq. 2} {/eq}:

$$\begin{align*} 8 \% x + 6 \% y &= 650 \\[0.3cm] 8 \% x + 6 \% \left( x + 1500 \right) &= 650 \\[0.3cm] \dfrac{8}{100} x + \dfrac{6}{100}\left( x + 1500 \right) &= 650 \\[0.3cm] 0.08 x + 0.06 \left( x + 1500 \right) &= 650 \\[0.3cm] 0.08 x + 0.06x + 0.06 \left( 1500 \right) &= 650 \\[0.3cm] 0.08 x + 0.06x + 90 &= 650 \\[0.3cm] 0.14 x + 90 &= 650 \\[0.3cm] 0.14 x &= 650 - 90 \\[0.3cm] 0.14 x &= 560 \\[0.3cm] x &= \dfrac{560}{0.14} \\[0.3cm] \therefore x &= 4000 \end{align*} $$

Substitute {eq}x = 4000 {/eq} in the {eq}\text{Eq. 1} {/eq}:

$$\begin{align*} y &= x + 1500 \\[0.3cm] &= 4000 + 1500 \\[0.3cm] \therefore y &= 5500 \end{align*} $$

Hence, the hotel charge before tax in the first city was {eq}\color{blue}{\$ 4000} {/eq} and the hotel charge before tax in the second city was {eq}\color{blue}{\$ 5500} {/eq}.


Learn more about this topic:

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How to Solve a System of Linear Equations in Two Variables

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Chapter 6 / Lesson 3
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A system of linear equations in two variables is an important concept that is used in many math disciplines. Review a detailed explanation of these systems and explore how to solve them using the substitution method.


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