Problem 48 - Elementary Algebra

Question

For moving purposes, the Hendersons bought 25 cardboard boxes for \(\$ 97.50\). There were two kinds of boxes: the large ones cost \(\$ 7.50\) per box, and the small ones cost \(\$ 3\) per box. How many boxes of each kind did they buy?

Step-by-Step Solution

Answer

The Hendersons bought 5 large boxes and 20 small boxes.

Step 1: Define Variables

Let \( x \) be the number of large boxes, and \( y \) be the number of small boxes.

Step 2: Set Up Equations

We know the Hendersons bought 25 boxes in total. Therefore, we have the equation: \[ x + y = 25 \] We also know the total cost of the boxes was \( \$97.50 \). Therefore, the cost equation is: \[ 7.5x + 3y = 97.5 \]

Step 3: Solve for one variable

Using the first equation \( x + y = 25 \), we can express \( y \) in terms of \( x \): \[ y = 25 - x \]

Step 4: Substitute to Solve for x

Substitute \( y = 25 - x \) into the cost equation \( 7.5x + 3y = 97.5 \): \[ 7.5x + 3(25 - x) = 97.5 \] Simplify and solve for \( x \). \\[ 7.5x + 75 - 3x = 97.5 \] \[ 4.5x + 75 = 97.5 \] Subtract 75 from both sides: \[ 4.5x = 22.5 \] Now, divide by 4.5: \[ x = 5 \]

Step 5: Find the value of y

Now that we have \( x = 5 \), substitute it back into the equation for \( y \):\[ y = 25 - x = 25 - 5 \]\[ y = 20 \]

Step 6: Verify the solution

Verify by plugging \( x = 5 \) and \( y = 20 \) into the original equations.Total number of boxes: \[ x + y = 5 + 20 = 25 \] The total cost should be \(7.5x + 3y = 97.5\):\[ 7.5(5) + 3(20) = 37.5 + 60 = 97.5 \] Both conditions are satisfied, so the solution is correct.

Key Concepts

linear equations

Linear equations are mathematical expressions that represent straight-line relationships between variables. In their simplest form, they consist of variables with coefficients and a constant value. These equations are of the form: \[ ax + by = c \] where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. Linear equations can have one or more variables, and they are often used to model real-world scenarios, like our moving box problem.

• In a system of linear equations, we have two or more linear equations working together.
• Each equation describes a line in a plane, and the solution to the system is the point or points where these lines intersect.
• For example, in the Henderson problem, \( x + y = 25 \) and \( 7.5x + 3y = 97.5 \) are such equations.

Knowing how to work with linear equations helps us find unknown values in practical situations, like determining the numbers of large and small boxes purchased.

word problems

Word problems are a thoughtful way of applying mathematical concepts to real-world scenarios. They often require translating a narrative into equations that can be solved mathematically. For example, the Hendersons' moving box problem involves understanding the situation, identifying the unknowns, and establishing relationships between numbers.

• First, read the problem carefully and understand the context.
• Identify what you need to find, often represented as variables in the equations.
• Clearly note down any given quantities and their relationships.

In our exercise, the word problem involves determining the number of each type of box purchased. Recognizing that you need two equations from the total number of boxes and their total cost is a crucial step. Word problems challenge you to translate everyday language into structured mathematical models and help improve problem-solving skills.

solving equations

Solving equations, particularly linear equations, is a fundamental skill in algebra. It involves finding the values of variables that make the equation true. There are various methods to solve systems of linear equations, including substitution and elimination. In the Hendersons' problem, we use substitution to solve the system of equations.- **Substitution Method:** This involves solving one equation for one variable and then substituting that expression into the other equation. It can be helpful when one equation is already solved for one variable.- **Steps followed here:** - Solve for \( y \) in terms of \( x \) using the equation \( x + y = 25 \) to get \( y = 25 - x \). - Substitute \( y = 25 - x \) into \( 7.5x + 3y = 97.5 \) and solve for \( x \).Finding the solution that fits all given conditions verifies that the solution works and is correct. Equations can appear complex, but breaking them into clear steps makes solving them manageable and logical. Understanding this approach for solving systems of equations assists in tackling many problems in algebra and beyond.