Problem 7 - Algebra 2

Question

Graph each equation. Check your work. $$ y-3=-2 x $$

Step-by-Step Solution

Answer

The graph of the equation is a straight line with a slope of -2 and a y-intercept at (0,3).

Step 1: Rearrange the equation into slope-intercept form

To graph the equation, it's easiest to have it in y = mx + b form, where m is the slope, and b is the y-intercept. Start by adding 2x to both sides of the equation to get y on one side by itself.

Step 2: Write the equation in slope-intercept form

After rearranging, the equation is y = -2x + 3, where the slope m is -2 and the y-intercept b is 3.

Step 3: Plot the y-intercept on the graph

Start by plotting the y-intercept (0,3) on the graph. This is where the line will cross the y-axis.

Step 4: Use the slope to plot another point

From the y-intercept, use the slope to find another point on the line. The slope -2 means you go down 2 units and right 1 unit from the y-intercept to plot a second point.

Step 5: Draw the line

Draw a straight line through the two points. This is the graph of the equation y - 3 = -2x.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential for graphing. It is expressed as y = mx + b, where m represents the slope of the line, and b indicates the y-intercept, which is the point where the line crosses the y-axis.

Converting an equation into this form simplifies the graphing process. For the exercise y-3=-2x, the equation is first rearranged by adding 2x to both sides, resulting in y = -2x + 3. Now, the equation is in the form where the slope, m, is -2 and the y-intercept, b, is 3. It's clear to see that -2 indicates the steepness and the direction of the line, while 3 tells us where to start plotting on the y-axis.

Y-Intercept

The y-intercept is a fundamental concept when working with linear graphs. It is the point where the line meets the y-axis, and in the slope-intercept form, it is given by the b in y = mx + b. In our example, the y-intercept is 3, which we find by setting x to zero, resulting in y = 0. This gives us the coordinates (0, 3).

Plotting the y-intercept is always the first step after rearranging the equation. It is the starting point from which the rest of the line's path will be determined. In graphing our linear equation, after placing a point at (0, 3), the line's next points can be plotted using the slope, which provides direction and steepness.

Plotting Points

Once the equation is in slope-intercept form and you have identified the y-intercept, the next step is to plot points to form the line on a graph. Starting from the y-intercept, we can use the slope to find the next point.

For the equation y = -2x + 3, the slope is -2. This means for each step right on the x-axis, the corresponding movement on the y-axis will be two units down because a negative slope indicates a line that decreases from left to right. Thus, from point (0, 3), we move 1 unit right (positive x-direction) and 2 units down (negative y-direction) to get to the next point, which is (1, 1).

By plotting the initial and subsequent points and then drawing a straight line through them, the graph of the linear equation takes shape, allowing you to visualize the relationship described by the equation.