Problem 58 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \( y \) is \( \frac{-5}{2} \).

Step 1: Understand the formula

The formula given is a linear equation: \( y = \frac{1}{2}x - 3 \). This means that for any given value of \( x \), you can substitute it into the equation to find the corresponding value of \( y \).

Step 2: Substitute the value of \( x \)

The problem provides \( x = 1 \). Substitute 1 for \( x \) in the equation: \( y = \frac{1}{2}(1) - 3 \).

Step 3: Calculate \( \frac{1}{2} \times 1 \)

Multiply \( \frac{1}{2} \) by the given \( x \) value (1): \( \frac{1}{2} \times 1 = \frac{1}{2} \).

Step 4: Subtract 3

Subtract 3 from \( \frac{1}{2} \): \( y = \frac{1}{2} - 3 = \frac{1}{2} - \frac{6}{2} = \frac{-5}{2} \).

Key Concepts

Solving for y

When working with linear equations like the one in the exercise, the main goal is often to find the value of the dependent variable, which in this case is \(y\). The linear equation given is \( y = \frac{1}{2}x - 3 \). This equation is in the form \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.
To solve for \(y\), you'll substitute a specific value for \(x\) as provided in a problem. Begin by identifying the given value for \(x\). Here, \(x = 1\). Substitute \(1\) into the equation wherever \(x\) appears. Calculating the result after substituting provides you with the corresponding \(y\) value.
This example demonstrates the basic skill of isolating \(y\) by substituting known values. Following this approach helps in finding points on a line described by the equation.

Substitution Method

The substitution method is a fundamental technique in solving equations, especially linear ones. In this method, a given value replaces a variable to simplify and solve the equation. For this exercise, the equation \( y = \frac{1}{2}x - 3 \) was presented with \(x = 1\) given.

• First, substitute \(1\) for \(x\) in the equation. This simplifies the expression to only include known numbers and operations.
• After substitution, you're left with \( y = \frac{1}{2}(1) - 3 \).
• Simplify \( \frac{1}{2}(1) \) to \( \frac{1}{2} \), then perform the arithmetic operation for subtraction from \(3\).

By substituting values, we can transform an equation with variables into one with numbers that are straightforward to calculate. This process is crucial in algebra, helping to validate and find solutions.

Basic Algebra Concepts

Understanding basic algebra is vital for solving equations. At its core, algebra involves using variables to represent numbers and finding their values through operations indicated by the equation. Here are some key concepts:

• Variables: Represent unknown values. In the equation \( y = \frac{1}{2}x - 3 \), \(x\) and \(y\) are variables.
• Coefficients: Numbers that multiply variables. Here, \(\frac{1}{2}\) is the coefficient of \(x\).
• Constants: Fixed values. In the equation, \(-3\) is a constant subtracted from the expression.
• Linear Equations: Equations that graph as straight lines. The form \( y = mx + b \) depicts a linear relationship;

Basic algebra helps bridge the gap between simple arithmetic and complex mathematical concepts. By understanding these elements, students can solve linear problems methodically and with greater confidence.