Problem 60 - Prealgebra

Question

Use the formula \(y=\frac{1}{2} x-3\) to find \(y\) if: $$X=4$$

Step-by-Step Solution

Answer

The value of \(y\) when \(x = 4\) is \(-1\).

Step 1: Identify Given Variables

We are given the equation \(y = \frac{1}{2}x - 3\) and the value \(x = 4\). Our goal is to find the corresponding value of \(y\).

Step 2: Substitute the Value of \(x\) into the Equation

Replace \(x\) in the equation with the value 4. This gives us the equation \(y = \frac{1}{2}(4) - 3\).

Step 3: Simplify the Equation

Calculate \(\frac{1}{2} \times 4\), which equals 2. Now, substitute this into the equation, resulting in \(y = 2 - 3\).

Step 4: Solve for \(y\)

Subtract 3 from 2 to get \(y = -1\).

Key Concepts

Substitution Method

The substitution method is a crucial technique in solving systems of equations or individual equations. To understand this method, you simply need to replace a variable with its known value or expression. For instance, in the exercise, we were provided with the equation \( y = \frac{1}{2}x - 3 \) and asked to find \( y \) when \( x = 4 \). By substituting \( x \) with 4 in the given equation, the expression becomes \( y = \frac{1}{2}(4) - 3 \).

The substitution method helps:

• Simplify an equation by reducing the number of variables.
• Provide a straightforward path to finding the solution.

It is often the first step when dealing with equations and ensures that you can solve for unknown variables step by step.

Solving Equations

Solving equations involves finding the value of unknown variables that satisfy the equation. In our example, once the substitution is made, the equation \( y = \frac{1}{2}(4) - 3 \) simplifies to \( y = 2 - 3 \). The final step is to perform basic arithmetic operations to find \( y \).

Here’s how you proceed with solving an equation:

• Substitute known values into the equation.
• Simplify the expression by performing the arithmetic operations.
• Arrive at the solution that makes the equation true.

In our exercise, this leads us to a solution where \( y = -1 \). Solving equations is an essential part of algebra, allowing us to make sense of relationships expressed in the form of mathematical models.

Prealgebra Concepts

Prealgebra introduces foundational mathematics concepts that are crucial for future algebraic understanding. Concepts such as variables, equations, and basic arithmetic operations are emphasized.

In our exercise, the concept of variables is important, as we identify \( x \) and \( y \) as the main variables in the equation \( y = \frac{1}{2}x - 3 \). Prealgebra also focuses on understanding how to manipulate these variables through operations like substitution, addition, and subtraction.

Here are some key prealgebra concepts demonstrated in the exercise:

• Identifying variables: Recognizing \( x \) and \( y \) as symbols representing numbers.
• Arithmetic with fractions: Simplifying expressions like \( \frac{1}{2}(4) \).
• Basic solving process: Sequentially arriving at the solution by substitution and simplification.

Mastering these prealgebra concepts is vital for advancing in mathematics, as they create the groundwork for tackling more complex algebraic problems later on.