Problem 56 - College Algebra with Corequisite Support

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.

Step 1: Understand the Equation

The profit equation is given as \( p = 3x + 4y \). Here, \( p \) represents the total profit, \( x \) is the number of item one, and \( y \) is the number of item two.

Step 2: Substitute Known Values

We are given that \( p = 453 \) and \( x = 75 \). Substitute these values into the profit equation to find \( y \). Thus, the equation becomes:\[ 453 = 3(75) + 4y \]

Step 3: Calculate the Expression

First calculate \( 3 \times 75 \), which equals \( 225 \). Substitute this result back into the equation:\[ 453 = 225 + 4y \]

Step 4: Isolate the Variable \( y \)

Subtract \( 225 \) from both sides of the equation to solve for \( 4y \):\[ 453 - 225 = 4y \]This simplifies to \( 228 = 4y \).

Step 5: Solve for \( y \)

Divide both sides by \( 4 \) to find the value of \( y \):\[ y = \frac{228}{4} = 57 \]

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed. They are an essential part of algebra and often used to model real-world situations. A linear equation typically has two variables and follows the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. For example, in our profit equation \( p = 3x + 4y \), the variables are \( x \) and \( y \). Here, \( 3x + 4y \) is a linear combination of the variables. Such equations are called linear because if you plot all possible solutions, they form a straight line.

Linear equations are powerful tools because they can represent relationships between different quantities in practical scenarios like profit, cost, or distance. They help us understand how changing one variable affects the other.
Using simple algebraic techniques, you can manipulate linear equations to find the value of one variable if you know the others. This is crucial for solving problems where multiple conditions or constraints are given.

Solving for Variables

Solving for a variable means finding its value that makes an equation true. In the context of our problem, we used the profit equation to find the value of \( y \) when we already knew \( p \) and \( x \).

To solve for a variable in a linear equation, follow these steps:

• Substitute known values into the equation.
• Simplify the equation by performing arithmetic operations.
• Isolate the variable, often by using inverse operations like addition, subtraction, multiplication, or division.

In our exercise, after substituting the known values, we simplified the equation to \( 453 = 225 + 4y \). From here, we subtracted 225 from both sides to isolate terms involving \( y \), resulting in \( 228 = 4y \). Dividing both sides by 4 gave us the value of \( y = 57 \).

It is important to perform these steps carefully and check your work, as each operation must balance both sides of the equation. This ensures you have correctly isolated the variable and found its accurate value.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In algebra, they form the basis of equations, which we then manipulate to solve problems.

An algebraic expression like \( 3x + 4y \) consists of terms where \( 3x \) and \( 4y \) are separate units. Each term consists of a coefficient (a numerical factor) and a variable. Understanding how to work with these expressions is crucial for solving equations.

Using algebraic expressions, you can model a wide range of situations by describing relationships between different quantities. They can be substituted with values and simplified to provide meaningful solutions, as demonstrated in our problem. By substituting \( x = 75 \) and \( p = 453 \) into \( 3x + 4y \), we can find the value of \( y \).

Grasping the nature of algebraic expressions allows you to translate real-world problems into mathematical language, making them easier to analyze and solve. They are fundamental to developing a deeper understanding of algebra and its applications in various fields, such as economics, physics, and engineering.