Problem 20 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ y=\frac{6-x}{3} $$

Step-by-Step Solution

Answer

The slope (m) of the line is -1/3 and the y-intercept (b) is 2.

Step 1: Recognize the format

The provided equation y = (6 - x)/3 does not have a standard y = mx + b form. So, we need to convert it into this form. The constants m and b are the solutions to our problem. Hence, the most important step will be to successfully transform the original equation into the correct form.

Step 2: Rearrange the Equation

The rearrangement should result in the form of y = mx + b. Start by distributing the denominator in the numerator, giving a new version of the equation as: \(y = 2 - \frac{1}{3}x \).

Step 3: Identify the Slope and Y-intercept

Now that the equation is in the standard form y = mx + b, the coefficient of x represents a slope and the constant term represents the y-intercept. Therefore, the slope (m) of the line is \(-\frac{1}{3}\) and the y-intercept (b) is 2.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in algebra that involves plotting the line represented by an equation. When dealing with equations in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, it's pretty straightforward to draft the line on the coordinate plane.

Here's how you can graph such an equation more effectively:

• First, identify the y-intercept \((b)\). This number is where the line crosses the y-axis. For instance, in the equation \(y = 2 - \frac{1}{3}x\), the y-intercept is 2.
• Plot the y-intercept on the graph at \((0, b)\).
• Next, use the slope \(m\) to determine the rise over run on the graph. The slope \(-\frac{1}{3}\) means you move down 1 unit vertically for every 3 units you move to the right horizontally.
• Starting from the y-intercept, move in accordance with the slope and plot another point.
• Finally, draw a line through the points plotted, extending it across the graph.

By following these steps, you’ll be able to visualise and understand the behavior and properties of linear equations better.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to work with. In the context of converting linear equations to slope-intercept form, it involves isolating variables and constants using operations like addition, subtraction, multiplication, and division.

Let's explore how algebraic manipulation was used in our example:

• The given equation \(y = \frac{6-x}{3}\) needs to be transformed into the \(y = mx + b\) form. To do this, distribute the denominator into each term of the numerator.
• This step involves breaking down the fraction: \(y = \frac{6}{3} - \frac{1}{3}x\).
• By simplifying, you get \(y = 2 - \frac{1}{3}x\), which is now in slope-intercept form.

Algebraic manipulation requires careful handling of each step to ensure precision. With practice, these skills enable solutions to complex problems with confidence and clarity.

Identifying Slope and Y-intercept

Identifying the slope and y-intercept is crucial for understanding and graphing linear equations. In the form \(y = mx + b\), the slope \(m\) represents the steepness and direction of the line, while the y-intercept \(b\) indicates where the line crosses the y-axis.

Here’s how to identify these from an equation:

• Once the equation is in \(y = mx + b\) form, the coefficient of \(x\) is the slope \(m\). In our case, it's \(-\frac{1}{3}\), indicating the line decreases as it moves from left to right.
• The constant term \(b\) is the y-intercept. In this equation, it's 2, meaning the line crosses the y-axis at \((0, 2)\).

Understanding these components helps in predicting the line's behavior on the graph. It makes finding solutions and analyzing relationships in graph problems much simpler.