Problem 61 - Understanding Elementary Algebra with Geometry

Question

An electronics discount store wants to use up a credit of \(\$ 9,110\) with its supplier to order a shipment of VCRs and TVs. Each VCR costs \(\$ 125\) and each TV costs \(\$ 165\) (a) Let \(v\) represent the number of VCRs and \(t\) represent the number of TVs. Write an equation that reflects the given situation. (b) Sketch the graph of this relationship. Be sure to label the coordinate axes clearly. (c) If 28 VCRs are ordered, use the equation you obtained in part (a) to find the number of TVs.

Step-by-Step Solution

Answer

Equation: 125v + 165t = 9,110. With 28 VCRs, 34 TVs can be ordered.

Step 1 - Set Up the Equation

Identify the variables and their corresponding costs. Let the number of VCRs be denoted as \( v \) and the number of TVs as \( t \). The cost of each VCR is \$125, and the cost of each TV is \$165. Therefore, the equation representing the total cost is: \[ 125v + 165t = 9,110 \]

Step 2 - Sketch the Graph

To sketch the graph of the equation \( 125v + 165t = 9,110 \), convert the equation into intercept form. First, find the intercepts by setting one variable to zero and solving for the other variable. For the VCR intercept, set \( t = 0 \): \[ 125v = 9,110 \] \[ v = \frac{9,110}{125} = 72.88 \] For the TV intercept, set \( v = 0 \): \[ 165t = 9,110 \] \[ t = \frac{9,110}{165} = 55.21 \] Plot these intercepts on the graph: \( v \) on the x-axis and \( t \) on the y-axis. Draw the line passing through these points.

Step 3 - Find the Number of TVs

Given the number of VCRs \( v \) is 28, substitute \( v = 28 \) into the equation from Step 1: \[ 125(28) + 165t = 9,110 \] Solving for \( t \): \[ 3,500 + 165t = 9,110 \] \[ 165t = 9,110 - 3,500 \] \[ 165t = 5,610 \] \[ t = \frac{5,610}{165} = 34 \]

Key Concepts

Linear Equations

Linear equations are mathematical statements that show a linear relationship between two variables. In this exercise, the relationship between the number of VCRs and TVs bought by the electronics discount store is shown using linear equations. The general form of a linear equation with two variables is: \[ ax + by = c \]. Here, \( x \) and \( y \) are variables, and \( a \), \( b \), and \( c \) are constants. When we introduce the variables into our problem, we get: \[ 125v + 165t = 9,110 \]. This equation tells us how the total cost (\$9,110) is divided between VCRs and TVs, where each VCR costs \$125 and each TV costs \$165.

Graphing Linear Functions

Graphing a linear function helps us visually understand the relationship between the variables. For graphing the equation \( 125v + 165t = 9,110 \), you can find the x-intercept and y-intercept.

• To find the x-intercept (where \( t = 0 \)): \[ 125v = 9,110 \] \[ v = 72.88 \]
• To find the y-intercept (where \( v = 0 \)): \[ 165t = 9,110 \] \[ t = 55.21 \]

Next, plot these points on graph paper with \( v \) (number of VCRs) on the x-axis and \( t \) (number of TVs) on the y-axis. Draw a line passing through these intercepts. This line shows all possible combinations of VCRs and TVs that the store can buy with their credit of \$9,110.

Variable Substitution

Variable substitution is a technique to solve equations where you replace the variable with a known value. In this exercise, you already know the number of VCRs ordered, which is \( v = 28 \). Plug this into the equation to find the number of TVs: \[ 125(28) + 165t = 9,110 \] This turns into: \[ 3,500 + 165t = 9,110 \] Subtract \( 3,500 \) from both sides: \[ 165t = 5,610 \] Divide by 165: \[ t = \frac{5,610}{165} = 34 \]. This means the store can order 34 TVs along with 28 VCRs using their credit.

Word Problems

Word problems often describe real-world situations through text, requiring the translation of words into mathematical equations. Here, the word problem describes a financial arrangement: buying VCRs and TVs with a fixed amount of credit. To solve word problems effectively:

• Identify and define the variables (e.g., \( v \) for VCRs, \( t \) for TVs).
• Use key information from the problem to form an equation.
• Solve the equation step-by-step using algebraic methods or graphing techniques.

Breaking down the problem into smaller steps, as outlined in the exercise, helps in understanding the relationships and solving the problem correctly.