Problem 56 - College Algebra

Question

If the profit equation for a small business selling \(x\) number of item one and \(y\) number of item two is \(p=3 x+4 y,\) find the \(y\) value when \(p=\$ 453\) and \(x=75\)

Step-by-Step Solution

Answer

The value of \( y \) is 57 when \( p = 453 \) and \( x = 75 \).

Step 1: Set up the equation

We start with the given profit equation: \( p = 3x + 4y \). We know \( p = 453 \) and \( x = 75 \). Substitute these values into the equation: \( 453 = 3(75) + 4y \).

Step 2: Simplify the equation

Calculate the product \( 3 \times 75 = 225 \) and substitute it back into the equation: \( 453 = 225 + 4y \).

Step 3: Isolate the term with y

Subtract 225 from both sides of the equation to get \( 453 - 225 = 4y \). Simplify the left side: \( 228 = 4y \).

Step 4: Solve for y

Divide both sides by 4 to solve for \( y \): \( 228 \div 4 = y \). This gives us \( y = 57 \).

Key Concepts

Profit Calculation

Profit calculation is a crucial aspect of any business, as it helps in determining the financial health and success of the venture. In simplistic terms, profit can often be expressed as a mathematical equation, where certain fixed or variable factors determine the total profit. In our exercise, the profit equation is given by \( p = 3x + 4y \). This equation indicates how the profits from selling two different items — item one and item two — contribute to the overall profit.

• Here, \( 3x \) represents the profit from selling \( x \) units of item one.
• The term \( 4y \) stands for the profit from \( y \) units of item two.

When solved, this tells us how variations in the number of items sold affect the total profit. The goal in profit calculation is to maximize the profit given certain constraints, such as price, cost of goods, and selling efforts.

Solving for Variables

Solving for variables is an essential skill in algebra that allows us to find unknown values using equations. It involves rearranging terms and isolating the variable of interest to find its value. Often, we are given certain numerical details within the problem, which we can use as substitutes in the equation. For example, in our problem, we know:

• The total profit \( p = 453 \).
• The number of \( x \) items sold, \( x = 75 \).

Using these pieces of information, we start by substituting the known values into the profit equation to solve for the unknown variable, \( y \). That means we insert \( p = 453 \) and \( x = 75 \) into \( p = 3x + 4y \). Hence, the equation becomes: \[ 453 = 3(75) + 4y \]The subsequent steps involve algebraic manipulation to isolate and solve for \( y \).

Algebraic Substitution

Algebraic substitution is a method in mathematics where we replace variables in an equation with given numbers or expressions to make solving the equation easier. This step helps in simplifying complex equations and, eventually, finding the solution more swiftly. In our current problem:

• The profit equation \( p = 3x + 4y \) has known values substituted into it, specifically \( p = 453 \) and \( x = 75 \).

The substitution process transforms the equation into one with only one variable: \[ 453 = 3(75) + 4y \]We calculate the product \( 3 \times 75 = 225 \), allowing us to rewrite the equation as:\[ 453 = 225 + 4y \]This substitution simplifies the task of isolating \( y \) so that further calculations lead us to determine its value. The final step involves solving the simplified equation by isolating \( y \) to find it equals 57. Proper application of substitution tends to make seemingly complex problems much more manageable.