Problem 24 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y=\frac{2}{3} x-2 $$

Step-by-Step Solution

Answer

Plot (0, -2); slope 2/3; line through (0, -2) and (3, 0).

Step 1: Understand the Equation

The given equation is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = \frac{2}{3} \) and \( b = -2 \).

Step 2: Plot the y-intercept

The y-intercept \( b \) is -2. This means that the line crosses the y-axis at the point (0, -2). Plot the point (0, -2) on the graph.

Step 3: Use the Slope to Find Another Point

The slope \( m = \frac{2}{3} \) indicates that for every 3 units you move to the right (positive x-direction), you move up 2 units (positive y-direction). From point (0, -2), count 3 units to the right and 2 units up to reach the point (3, 0). Plot this point.

Step 4: Draw the Line

Use a ruler to draw a straight line through the points (0, -2) and (3, 0). This line represents the graph of the equation \( y = \frac{2}{3} x - 2 \).

Key Concepts

Slope-Intercept Form

Linear equations are often expressed in what we know as the slope-intercept form. This form is designed to make graphing and understanding linear relationships easy. A typical equation in this format looks like this: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) stands for the y-intercept. This setup provides an intuitive way to picture the line just by looking at these two components.

• The **slope** tells you how steep the line is or how it is inclined.
• The **y-intercept** tells you precisely where the line will intersect the y-axis.

In the example equation \( y = \frac{2}{3}x - 2 \), the equation is already in the slope-intercept form. This setup makes it straightforward to quickly pinpoint and plot the line on a graph.

Y-intercept

The y-intercept is a fundamental part of graphing linear equations. It refers to the exact spot on the graph where the line crosses the y-axis. In mathematical terms, it's the point \((0, b)\), where \(b\) is the y-intercept value.

For the equation \( y = \frac{2}{3}x - 2 \), the y-intercept is \(-2\). This means that if you start at the origin, where the x and y axes meet (point \((0, 0)\)), and move directly down on the y-axis, you'll find the line crossing at \((0, -2)\). By plotting this point, you have a clear starting position to define the line's path, giving a foundational anchor to map out the rest of the line.

Slope of a Line

The slope of a line is a measure of how the line angles or tilts within the graph. It's described numerically as the rise over the run, a formula encapsulated as \( m = \frac{\text{rise}}{\text{run}} \). This means it’s the number of units the line goes up (rise) for every number of units it moves horizontally to the right (run).

In our equation \( y = \frac{2}{3}x - 2 \), the slope is \( \frac{2}{3} \). It indicates that for every 3 units you move to the right, you ascend 2 units upwards. Begin plotting by starting at the y-intercept (0, -2), and from there, move 3 units horizontally to the right and 2 units up to find your second point, (3, 0). This gives you two critical points needed to draw the line accurately and highlights how the slope dictates the direction and angle of the line.