Problem 16 - Elementary Algebra

Question

Graph the equations. $$ -2 x+3 y=-6 $$

Step-by-Step Solution

Answer

Answer: The slope (m) of the given equation is 2/3 and the y-intercept (b) is -2. To graph the line, plot the y-intercept (0, -2) on the graph. Use the slope to find additional points by moving 3 units to the right and 2 units up from the y-intercept. Join these points to draw the line representing the given equation.

Step 1: Convert the equation to slope-intercept form (y = mx + b)

To do this, we will isolate y by following these steps: 1. Add 2x to both sides of the equation. 2. Divide all terms by 3. $$ -2 x+3 y=-6 \\ 2x - 2x + 3y = 2x - 6 \\ 3y = 2x - 6 \\ y = \frac{2x}{3} - 2 $$ Now the equation is in the slope-intercept form: $$y = \frac{2}{3}x - 2$$.

Step 2: Identify the slope and y-intercept

In the equation $$y = \frac{2}{3}x - 2$$, the slope (m) is $$\frac{2}{3}$$, and the y-intercept (b) is $$-2$$.

Step 3: Plot the y-intercept

As the y-intercept is -2, we will plot the point (0, -2) on the graph.

Step 4: Use the slope to find additional points

The slope of the line is $$\frac{2}{3}$$, which means that for every 3 units we move to the right on the x-axis (positive direction), we move 2 units up on the y-axis. Starting at the y-intercept (0, -2), move 3 units to the right and 2 units up to find a new point, (3, 0). You can repeat this process to find more points if needed.

Step 5: Draw the line

Using the two points, (0, -2) and (3, 0), on the graph, draw a straight line. The result is the graph of the given linear equation. And now, the graphical representation of the given equation is complete.

Key Concepts

Graphing Equations

Graphing a linear equation allows you to visualize its behavior on a coordinate plane. It's like taking a mathematical concept and drawing it out for clarity. For a linear equation, we need two main components: the slope and the y-intercept.
Identifying these parts will enable us to plot points that belong to the line represented by the equation. Once we have enough points, we can draw a straight line through them, representing the complete set of solutions for the equation.

• Start with a clear equation format: We typically use the slope-intercept form.
• Identify key elements like slope and y-intercept from the equation.
• Plot these components accurately on a coordinate grid to draw the graph.

Graphing is an essential skill because it provides a visual perspective of an equation's solutions, allowing us to see patterns and make predictions.

Slope-Intercept Form

The slope-intercept form is a very handy way to write linear equations. It is written as \( y = mx + b \), where:

• \( m \) represents the slope of the line
• \( b \) is the y-intercept, the point where the line crosses the y-axis

Using this form is convenient because it immediately tells us the slope and the y-intercept.
When an equation is given, converting it to this form can simplify the process of graphing, as you can directly extract all necessary information.
For example, in the equation \( y = \frac{2}{3}x - 2 \), the slope is \( \frac{2}{3} \) and the y-intercept is \( -2 \), making it easy to plot and draw the line.

Y-Intercept

The y-intercept is a crucial component of a linear equation as it marks where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) signifies the y-intercept.
This point is extremely valuable for graphing because it offers a specific starting point for plotting the line on a coordinate plane. In practical terms, you place a dot on the y-axis at \( (0, b) \).

• This step simplifies the process of plotting since every line only has one y-intercept.
• For instance, for \( y = \frac{2}{3}x - 2 \), the y-intercept is at \( -2 \), so you start by plotting the point \( (0, -2) \).

This concrete start point allows you to use the slope to find additional points, making the graphing much easier.

Finding Slope

The slope of a line measures its steepness and direction. It is denoted as \( m \) in the slope-intercept form \( y = mx + b \). The slope is calculated as the 'rise over run', which describes how much the line goes up (rise) for a certain amount it moves to the right (run).
A positive slope means the line is going upward, while a negative slope means it is going downward.

• For example, a slope of \( \frac{2}{3} \) indicates that for every 3 units moved horizontally, the line rises 2 units vertically.
• Starting from the y-intercept, use the slope to find subsequent points by moving along the grid.

Understanding slope is vital because it gives the line its shape and can affect how variables in the equation interact. In practical terms, knowing the slope helps you efficiently and accurately plot additional points needed for drawing the graph of an equation.