Problem 22 - Intermediate Algebra

Question

Salaries. \(\quad\) A student working for a delivery company earns S57.50 per day plus \(\$ 4.75\) for each package she delivers. How many deliveries must she make each day to earn \(\$ 200\) a day?

Step-by-Step Solution

Answer

She must deliver 30 packages per day to earn $200.

Step 1: Define Variables and Equation

Let \( x \) represent the number of packages delivered in a day. The total earnings in a day can be represented by the equation: \( 57.50 + 4.75x = 200 \), where \( 57.50 \) is the fixed daily earning, and \( 4.75x \) is the earning from delivering \( x \) packages.

Step 2: Isolate the Variable Term

Subtract \( 57.50 \) from both sides of the equation to isolate the term with \( x \):\[4.75x = 200 - 57.50\]\[4.75x = 142.50\]

Step 3: Solve for x

To find \( x \), divide both sides by \( 4.75 \):\[x = \frac{142.50}{4.75}\]Calculate the division to find \( x \):\[x = 30\]This indicates that the student needs to deliver 30 packages.

Key Concepts

Understanding Linear Equations

Linear equations form the bedrock of solving many algebra word problems. A linear equation is an equation that makes a straight line when graphed on a coordinate plane. It usually takes the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. This type of equation describes a proportional relationship between the variables.
In our problem, the linear equation is \( 57.50 + 4.75x = 200 \). Here, it connects the fixed daily earnings and earnings per package to the total earnings goal of \( \$200 \). Understand that the equation represents the student's daily earnings, which can be controlled by modifying the number of packages delivered, denoted by \( x \).
Linear equations are essential because they help solve for unknown variables, like deciding how many packages need to be delivered to achieve a certain earning.

The Concept of Variable Isolation

Variable isolation is a crucial step in solving linear equations. It involves manipulating the equation so that the variable of interest is on one side of the equation by itself. In simpler terms, you isolate the variable to understand its value clearly.
Consider the equation from our problem: \( 57.50 + 4.75x = 200 \). We start isolating the variable \( x \) by performing operations that simplify the equation. The first operation is subtracting \( 57.50 \) from both sides, giving us \( 4.75x = 142.50 \).

• This process maintains the equality of the equation because we apply the same operation on both sides.
• Doing this helps to focus solely on the packages' contribution to total earnings.

After isolating \( 4.75x \), you are closer to finding the exact number of packages needed.

Techniques for Solving Equations

Once we have isolated the variable term, the next step is solving the equation. Solving involves finding the actual number that the variable stands for. Continue from our isolated equation: \( 4.75x = 142.50 \).

• Divide both sides of the equation by \( 4.75 \) to solve for \( x \).
• This step transforms the equation into \( x = \frac{142.50}{4.75} \), simplifying it further to \( x = 30 \).

Now, you see that \( x \) equals 30, meaning the student needs to deliver 30 packages to earn \( \$200 \) a day.
Solving equations often requires performing reverse operations to undo the arithmetic involving the variable. Understanding this concept enables you to apply similar techniques to other algebra problems. This method ensures clarity and accuracy when working through mathematical texts.