Problem 89 - Algebra for College Students

Question

Solve each problem. An office manager is placing an order for note pads at \(\$ 1\) each and binders at \(\$ 2\) each. The total cost of the order must be \(\$ 100 .\) Write an equation for the total cost and graph it. If he orders 30 note pads, then how many binders must he order?

Step-by-Step Solution

Answer

He must order 35 binders.

Step 1: Define the variables

Let the number of note pads be denoted by \( x \) and the number of binders be denoted by \( y \).

Step 2: Write the cost equation

Each note pad costs \( \$ 1 \) and each binder costs \( \$ 2 \). The total cost is \( \$ 100 \). Therefore, the cost equation is: \[ x + 2y = 100 \]

Step 3: Solve for binders when note pads are given

Substitute \( x = 30 \) into the equation and solve for \( y \): \[ 30 + 2y = 100 \] \[ 2y = 70 \] \[ y = 35 \]

Step 4: Graph the equation

To graph the equation \( x + 2y = 100 \), find the intercepts. For the x-intercept, set \( y = 0 \): \[ x = 100 \] For the y-intercept, set \( x = 0 \): \[ 2y = 100 \] \[ y = 50 \] Plot these points \((100, 0)\) and \((0, 50)\) and draw a line through them.

Key Concepts

Graphing Equations

Graphing equations helps you visualize relationships between variables. In this exercise, we have the equation \[ x + 2y = 100 \] which shows the total cost of note pads and binders. To graph it, you need two key points: the x-intercept and the y-intercept. For the x-intercept, set y to 0 and solve for x: \[ x = 100 \] This gives the point \( (100, 0) \). For the y-intercept, set x to 0 and solve for y: \[ 2y = 100 \] This simplifies to \[ y = 50 \] giving us the point \( (0, 50) \). Plot these points on the graph. Then, draw a straight line through them. This line represents all the possible combinations of note pads and binders that total \$100\. Graphing this line helps you see all the combinations visually.

Solving Linear Equations

Solving linear equations is about finding the values of variables that make the equation true. Here, we start with the cost equation for note pads and binders \( x + 2y = 100 \). Suppose we know the number of note pads \( x = 30 \). Substitute \( x \) into the equation \[ 30 + 2y = 100 \]. The next step is to simplify and solve for \( y \). First, subtract 30 from both sides: \[ 2y = 70 \]. Divide by 2: \[ y = 35 \]. This tells us that if the office manager orders 30 note pads, he must also order 35 binders. It's crucial to follow these steps in solving any linear equation:

• Identify the variables
• Set up the equation
• Substitute known values
• Simplify and solve for the unknown variable

Consistency and practice make solving linear equations easier and faster!

Intercepts

Intercepts are points where the graph crosses the axes. They are essential in graphing linear equations. In our example, the equation \( x + 2y = 100 \) has two intercepts:

• The x-intercept is where the line crosses the x-axis. To find it, set \( y = 0 \) and solve for \( x \): \[ x = 100 \] giving the point \( (100, 0) \).
• The y-intercept is where the line crosses the y-axis. Set \( x = 0 \) and solve for \( y \): \[ 2y = 100 \] which simplifies to \[ y = 50 \] giving the point \( (0, 50) \).

Intercepts provide easy reference points for plotting the graph. Just plot these points, and draw a line through them to visualize the equation. Understanding intercepts is key in graphing equations quickly and accurately.