Problem 30 - Prealgebra

Question

PHONE CARDS A telephone calling card allows for 25 ç per minute plus a one- time service charge of \(75 \mathrm{e}\). If the total cost of the card is \(\$ 5,\) solve \(25 m+75=500\) to find the number of minutes you can use the card.

Step-by-Step Solution

Answer

You can use the card for 17 minutes.

Step 1: Understanding the problem

We need to find the number of minutes, \(m\), that can be used on the phone card given the total cost is \(\$5\), or \(500 \text{ç}\). The equation given is \(25m + 75 = 500\). Our goal is to solve for \(m\).

Step 2: Isolating the variable term

Subtract the one-time service charge from both sides of the equation to isolate the term with \(m\): \[25m + 75 - 75 = 500 - 75\]This simplifies to:\[25m = 425\]

Step 3: Solving for the variable

Divide both sides of the equation by 25 to solve for \(m\): \[m = \frac{425}{25}\]\[m = 17\]

Step 4: Conclusion

The number of minutes you can use the card is 17 minutes.

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they contain no exponents higher than one. They are one of the simplest forms of equations and often appear in the format of \(ax + b = c\). This format simply represents a straight line on a graph, hence the term "linear."

The exercise you're working with is a perfect example. The equation \(25m + 75 = 500\) is a linear equation, where \(m\) is the variable representing the number of minutes. Linear equations typically involve one or two variables only, and they're all about finding the value(s) that make the equation true.

Solving them frequently involves straightforward arithmetic operations like addition, subtraction, multiplication, and division. Recognizing these equations will make them easier to handle and solve.

Isolation of Variables

To solve an equation, you often need to "isolate" the variable. This means getting the variable on one side of the equation on its own, without any other numbers attached to it. In our example, we're solving for \(m\), the number of minutes.

Isolating \(m\) involves moving other numbers to the opposite side of the equation. Essentially, you are peeling away unwanted numbers from around \(m\) to find its value. Each step should simplify the equation, bringing us closer to the truth that \(m\) represents.

Subtraction in Equations

Subtraction is a key operation when solving equations. In the context of equations like the one you're dealing with, subtraction is used to eliminate constants from one side of the equation.

For the equation \(25m + 75 = 500\), we subtract 75 from both sides. This action "cancels out" the 75 on the left, simplifying it to \(25m = 425\). The subtraction ensures that only terms involving our variable, \(m\), are left for further manipulation.

Understanding subtraction's role is crucial in working out the logical sequence of steps needed to isolate and solve for variables efficiently.

Division in Equations

Once you've isolated the variable term using addition or subtraction, division is often the next step to finally solve for the variable. This is especially true when you have an equation like \(ax = c\), where you need to find \(x\).

In the linear equation \(25m = 425\), the goal is to determine what single value for \(m\) satisfies this equation. By dividing both sides by 25, we simplify the equation further: \(m = \frac{425}{25}\).

The division reduces the coefficient of \(m\) from 25 to 1, thereby solving for \(m\) and providing the solution \(m = 17\). Division is a vital step that converts these manipulated expressions into final answers that are easy to interpret.