Problem 10 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A discount building supplies store sells both first-quality and second-quality floor tiles. Robin buys three cases of first-quality tiles and one case of second-quality tiles for a total of \(\$ 66,\) while Gene buys one case of first-quality tiles and three cases of second-quality tiles for a total of \(\$ 54 .\) What is the cost per case for each type of tile?

Step-by-Step Solution

Answer

The cost per case of first-quality tiles is 18 dollars and the cost per case of second-quality tiles is 12 dollars.

Step 1: Define Variables

Let the cost per case of first-quality tiles be \( x \) dollars and the cost per case of second-quality tiles be \( y \) dollars.

Step 2: Set Up Equations

Use the information given in the problem to set up two equations. For Robin: 3 cases of first-quality tiles + 1 case of second-quality tiles = 66 dollars So, the equation is: \[ 3x + y = 66 \] For Gene: 1 case of first-quality tiles + 3 cases of second-quality tiles = 54 dollars So, the equation is: \[ x + 3y = 54 \]

Step 3: Solve the System of Equations - Elimination Method

Multiply the second equation by 3 to align the coefficients of \( x \): \[ 3(x + 3y) = 3(54) \] Which results in: \[ 3x + 9y = 162 \] Now, subtract the first equation from this new equation: \[ (3x + 9y) - (3x + y) = 162 - 66 \] This simplifies to: \[ 8y = 96 \] Solve for \( y \): \[ y = 12 \]

Step 4: Solve for \( x \)

Substitute \( y = 12 \) back into the second original equation: \[ x + 3(12) = 54 \] This simplifies to: \[ x + 36 = 54 \] Solve for \( x \): \[ x = 18 \]

Step 5: Conclusion

The cost per case of first-quality tiles is \( 18 \) dollars and the cost per case of second-quality tiles is \( 12 \) dollars.

Key Concepts

Systems of Equations

When a problem involves multiple variables and relationships between them, we often use systems of equations to solve it. In this exercise, we need to find the cost of first-quality and second-quality tiles. Two transactions are given, each representing a unique equation. By solving these equations together, we can find the cost per case for each type of tile. Systems of equations allow us to solve for multiple variables simultaneously, making it a powerful tool in algebra. They can be solved using several methods, like substitution, elimination, or graphical methods, but here we'll focus on the elimination method.

Variable Definition

The first step in solving a word problem is defining the variables. Here, we define two variables:

• Let x represent the cost per case of first-quality tiles.
• Let y represent the cost per case of second-quality tiles.

Clearly defining variables helps in translating the word problem into mathematical equations. It simplifies the problem and prepares us for setting up the necessary equations. Always ensure that your variables are related to what you need to find in the problem.

Elimination Method

The elimination method is a systematic way of removing one variable to solve for the other. Let's recount the steps:

• We have two equations from the problem:
  • Equation 1:
    3x + y = 66
  • Equation 2: x + 3y = 54
• Multiply Equation 2 by 3:
  3(x + 3y) = 3(54)
  Which simplifies to:
  3x + 9y = 162
• Subtract Equation 1 from this new equation:
  (3x + 9y) - (3x + y) = 162 - 66
  This results in: 8y = 96
• Solve for y:
  y = 12

By eliminating x, we have effectively found the value of y. This method is particularly useful for linear equations and helps in simplifying the solving process.

Algebraic Solution Steps

Once we've used the elimination method to find one variable, we can substitute it back into one of the original equations to find the other variable. Here are the steps:

• Substitute y = 12 into the second original equation:
  x + 3(12) = 54
  This simplifies to: x + 36 = 54
• Solve for x:
  x = 18

Finally, we have both variables:

• x = 18, the cost per case of first-quality tiles
• y = 12, the cost per case of second-quality tiles

By following structured algebraic steps, we ensure accuracy and clarity in solving word problems. Always double-check your calculations to avoid errors.