Question

Sketch the line through the given point with the indicated slope. $$ (2,3) ; \quad-2 $$

Short answer

The equation of the line through the point (2, 3) with a slope of -2 is \(y = -2x + 7\). To sketch the line, plot the given point (2, 3) and the y-intercept (0, 7), and then draw a straight line passing through these points.

Step-by-step solution

Write down the given point and slope

We are given the point (2, 3) and the slope, m = -2.

Use the slope-intercept form

We know that the slope-intercept form of a line equation is \(y = mx + b\), where m is the slope, and b is the y-intercept. We have the slope, so now we need to find the value of b. We can substitute the given point (2, 3) into the equation to do this.

Substitute the point and slope into the equation

Plug the point (2, 3) and the slope -2 into the equation \(y = mx + b\): \(3 = -2(2) + b\)

Solve for b

Solve the equation for b: \(3 = -4 + b\) \(b = 7\)

Write the equation of the line

Now that we have the slope and the y-intercept, we can write the equation of the line: \(y = -2x + 7\)

Sketch the line

To sketch the line, first plot the point (2, 3) on a graph. Next, plot the y-intercept (0, 7). Now that we have two points, draw a straight line passing through both points to represent the line \(y = -2x + 7\). With this sketched line, we have successfully plotted the line through the given point (2, 3) with the indicated slope of -2.

Key concepts

Slope-Intercept Form

Understanding the structure of the slope-intercept form is essential when it comes to graphing linear equations. The slope-intercept form of a line is written as (y = mx + b), where m represents the slope of the line, and b represents the y-intercept. This format is particularly useful because it gives us straightforward information about the graph of the line.

In our exercise, we're dealing with the slope-intercept form and graphing a line through a specific point with a given slope. By knowing the slope and one point, we can eventually find the y-intercept and then draw the entire line. The process is straightforward: we plug the known point and slope into the formula, solve for b, and voilà, we've got the equation that will guide us to sketch the line accurately on a graph.

Y-Intercept

The y-intercept is an important concept when graphing linear equations. It is the point where the line crosses the y-axis, marked by the coordinates (0, b) in the slope-intercept equation (y = mx + b).

The y-intercept adds a starting point for drawing our line on the graph. In our solution, after finding the slope, we used the given point (2, 3) to solve for the y-intercept, which we found to be 7. Thus, the y-intercept of our line is at the point (0, 7). Plotting the y-intercept on the graph gives us a foundation that, along with another point on the line, helps us in drawing the line smoothly and accurately.

Slope of a Line

The slope of a line, referred to as m in the slope-intercept formula, is a measure of how steep the line is and the direction it travels across the coordinate plane. Slope is calculated as the ratio of the vertical change between two points on the line (rise) to the horizontal change (run).

In our exercise, the slope is given as -2. This negative slope tells us that for every one unit we move to the right on the x-axis, we must move two units down, which results in a descending line from left to right. Slope is critical because it defines the line's angle and direction; in our case, the steepness and downward direction were key to graphing the line correctly.