Problem 62 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \(y\) is \(-4\).

Step 1: Identify the given values

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

Step 2: Substitute the known value of x into the formula

Replace \(x\) in the formula with \(-2\). The equation becomes: \(y = \frac{1}{2}(-2) - 3\).

Step 3: Perform the multiplication

Calculate \(\frac{1}{2} \times (-2)\). This results in \(-1\). Thus, the equation simplifies to: \(y = -1 - 3\).

Step 4: Simplify by performing the subtraction

Subtract \(3\) from \(-1\). Compute \(-1 - 3 = -4\). Hence, \(y = -4\).

Key Concepts

Linear Equations

Linear equations are fundamental in prealgebra. They represent relationships between variables, typically in the form of a straight line when graphed.
These equations have the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The slope \( m \) shows how much \( y \) changes for a unit change in \( x \). If you see an equation like \( y = \frac{1}{2}x - 3 \), it's telling you that for every change in \( x \), \( y \) changes by half that amount, then decreases by 3.
To find \( y \) for any given \( x \), you simply perform the arithmetic operations as the equation instructs. This process shows the importance of understanding each part of the equation. It helps you predict and calculate the outcomes by using any given value of \( x \). Linear equations are powerful tools for describing and predicting real-world scenarios in simple terms.

Substitution Method

The substitution method is a key technique in solving equations. It simplifies the problem by replacing one variable with its given or known value.
Imagine you have an equation like \( y = \frac{1}{2}x - 3 \) and you're told \( x = -2 \). Instead of balancing between \( x \) and \( y \), you substitute \( x \) with \(-2\), creating a straightforward equation: \( y = \frac{1}{2}(-2) - 3 \).
By using substitution:

• You focus on one variable at a time.
• You transform a complex equation into one that's easy to solve step-by-step.

This method is especially useful in systems of equations, where managing multiple variables simultaneously can be cumbersome. By mastering substitution, you can handle both simple and complex algebraic problems more efficiently.

Basic Arithmetic Operations

Basic arithmetic operations form the cornerstone of solving linear equations and using the substitution method.
These operations include addition, subtraction, multiplication, and division. For instance, in the equation \( y = \frac{1}{2}(-2) - 3 \), you need to:

• Multiply: \( \frac{1}{2} \times (-2) = -1 \).
• Subtract: \( -1 - 3 = -4 \).

Each step relies on your ability to correctly perform these simple arithmetic actions. It doesn’t just help in finding a solution but ensures accuracy, especially in multi-step problems.
Understanding each operation's role in an equation gives you confidence in solving them. It also reinforces the significance of order of operations, ensuring you perform multiplications and divisions before additions and subtractions. Remember, precise arithmetic keeps your math on track!