Problem 81 - Algebra 1

Question

GRAPHING Write the equation in slope-intercept form, and then graph the equation. Label the \(x\) - and \(y\) -intercepts on the graph. $$ -3 x+y+6=0 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = 3x + 6\). The x-intercept is (-2,0) and the y-intercept is (0,6).

Step 1 Convert to slope-intercept form

Rearrange the given equation to get \(y\) by itself on one side. Add 3x to both sides: \(y = 3x + 6\)

Step 2 Identify the slope and y-intercept

The coefficient of x in this equation is the slope \(m\), and the constant is the y-intercept \(b\). So, the slope \(m\) is 3 and the y-intercept \(b\) is 6.

Step 3 Find the x-intercept

To find the x-intercept, set \(y\) to 0 and solve for \(x\): \(0 = 3x + 6\). Solve for \(x\), \(x = -2\)

Step 4 Draw the graph

Plot the y-intercept (0,6). Then, use the slope to find another point. Starting from the y-intercept, move up 3 units (slope) and over to the right 1 unit, and place the second point. Repeat this process to get more points, if necessary. Draw a line through the points, extending in both directions.

Step 5 Label the intercepts

Label the x-intercept as (-2,0) and the y-intercept as (0,6).

Key Concepts

Understanding Slope-Intercept Form

The equation of a line in slope-intercept form is a simple and intuitive way to look at linear relationships. It's expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) indicates the y-intercept, the point where the line crosses the y-axis.

In the exercise provided, after rearranging \(-3x + y + 6 = 0\), we get \(y = 3x + 6\), which is now in slope-intercept form. Here, the slope \(m\) is 3, meaning for every unit we move horizontally to the right on the graph, we will move upward by 3 units. The y-intercept \(b\) is 6, showing where the line crosses the y-axis; at the point (0, 6). Think of the slope as the line's steepness or incline, and the y-intercept as your starting point if you walked along the line.

Understanding the slope-intercept form can greatly simplify graphing, because with just these two pieces of information - the slope and y-intercept - one can proceed to graph the entire line.

Finding X-Intercept and Y-Intercept

Intercepts offer a straightforward way to sketch a linear equation. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis. To find the y-intercept as you've seen in our exercise, we need only look at the 'b' value in the slope-intercept form; it's where the line touches the y-axis when \(x = 0\).

To determine the x-intercept, where the line crosses the x-axis, we set \(y = 0\) in the equation and solve for \(x\). In our exercise, doing so gives us the equation \(0 = 3x + 6\), leading us to find an x-intercept at \(x = -2\), or the point (-2, 0). It's essential to understand that a line's intercepts are key elements in quickly plotting its course on a graph, serving as anchor points from which the rest of the line can be drawn.

Plotting Points on a Graph

Graphing is a visual representation of mathematical concepts. When plotting points on a graph, we use an ordered pair \((x, y)\), which tells us the precise location of that point. The first number, \(x\), informs us how far to the left or right of the origin (0, 0) the point is, and the second number, \(y\), indicates how far up or down that point is situated.

As you begin plotting, always start with the y-intercept, here (0, 6), the first clear point on the line. Next, use the slope to find the next point. In the exercise, the slope is 3, representing the ratio of the rise over run, so from the y-intercept, we go up 3 units and over 1 unit right to find the next point. You can repeat this process to plot additional points.

Ensure your points form a straight line and extend it in both directions, capturing the full range of the linear relationship. By mastering point plotting, you will find interpreting data and relationships in graph form much simpler.