Problem 24 - 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. An artist's supply store sold a total of 20 canvases for \(\$ 172\). If some of the canvases cost \(\$ 7.50\) each and the remainder cost \(\$ 10.25\) each, how many of each type were sold?

Step-by-Step Solution

Answer

12 canvases were sold at \(\text{\textdollar7.50}\) each and 8 at \(\text{\textdollar10.25}\) each.

Step 1: Define variables

Let \(x\) be the number of canvases that cost \(\text{\$7.50}\) each and \(y\) be the number of canvases that cost \(\text{\$10.25}\) each.

Step 2: Set up equations

We know that the total number of canvases sold is 20. Thus, the equation is: \[ x + y = 20 \] The total cost of the canvases is \(\text{\$172}\). Therefore, the equation for the total cost is: \[ 7.50x + 10.25y = 172 \]

Step 3: Solve the first equation for one variable

Solve the first equation for \(y\): \[ y = 20 - x \]

Step 4: Substitute into the second equation

Substitute \(y = 20 - x\) into the second equation: \[ 7.50x + 10.25(20 - x) = 172 \]

Step 5: Simplify and solve

Simplify the equation: \[ 7.50x + 205 - 10.25x = 172 \] Combine like terms: \[ -2.75x + 205 = 172 \] Subtract 205 from both sides: \[ -2.75x = -33 \] Divide both sides by -2.75: \[ x = 12 \]

Step 6: Find the other variable

Substitute \(x = 12\) back into the equation \(y = 20 - x\): \[ y = 20 - 12 \] So, \[ y = 8 \]

Step 7: State the solution

There were 12 canvases sold at \(\text{\$7.50}\) each and 8 canvases sold at \(\text{\$10.25}\) each.

Key Concepts

System of Equations

A System of Equations is a set of two or more equations with the same variables. They are useful for solving problems where multiple conditions must be satisfied simultaneously. In the given exercise, we have two conditions:
1. The total number of canvases sold is 20.
2. The total cost of the canvases is \$172.\ When we translate these conditions into mathematical equations, we get:
\( x + y = 20 \)
\( 7.50x + 10.25y = 172 \)
Here, the variables \( x \) and \( y \) represent the number of canvases sold at each price. These equations form our System of Equations. To solve such a system, you need techniques like substitution or elimination methods. By solving these equations, we can find the values of \( x \) and \( y \) that meet both conditions.

Verbal Problems

Verbal Problems, often seen in algebra, require you to translate words into mathematical equations. This translation involves identifying the knowns and unknowns and setting up equations based on the information given. In our problem, we're told:
1. A total of 20 canvases were sold.
2. The total revenue was \$172.\
3. Some canvases cost \$7.50\ each, and others cost \$10.25\ each.
We translate these statements into variables and equations. Let \( x \) be the number of canvases at \$7.50\ and \( y \) be the number at \$10.25\. Thus, the equations are:
\( x + y = 20 \)
\( 7.50x + 10.25y = 172 \)
This step is crucial because it lays the foundation for solving the problem algebraically. Always ensure you understand the problem before setting up equations. This helps in avoiding mistakes and ensures you accurately translate the verbal problem into a solvable mathematical form.

Substitution Method

The Substitution Method is a way to solve systems of equations, particularly useful when one equation is easily solvable for one variable. Here’s how to use it effectively:
Step 1: Solve one of the equations for one variable. In our case, solving \( x + y = 20 \) for \( y \):
\( y = 20 - x \)
Step 2: Substitute this expression into the other equation. Plug \( y = 20 - x \) in the cost equation:
\( 7.50x + 10.25(20 - x) = 172 \)
Step 3: Simplify and solve for the remaining variable. Distribute and then combine like terms:
\( 7.50x + 205 - 10.25x = 172 \)
\( -2.75x + 205 = 172 \)
Subtract 205 from both sides and solve for \( x \):
\( -2.75x = -33 \)
\( x = 12 \)
Step 4: Substitute \( x \) back to find \( y \). Using \( y = 20 - x \):
\( y = 20 - 12 \)
\( y = 8 \)
Thus, the Substitution Method simplifies finding solutions and ensures that both original conditions are satisfied by the solution.