Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((1,3)\) \(m=3\)

Short answer

Given the point and slope, the equation of the line is \(y = 3x\). The line passes through the point \((1,3)\) and also through the origin \((0,0)\), rising to the right with a slope of \(3\).

Step-by-step solution

Identify given point and slope

The given point that the line passes through is \((1,3)\), and the slope \(m\) of the line is provided as \(3\).

Substitute into the slope-intercept formula

We use the point-slope form of the equation. Start with \(y-y1=m(x-x1)\), and substitute the given point \(\((1,3)\)\) and slope \(3\) to get the equation of the line. This leads to \(y - 3 = 3(x - 1)\).

Simplify the equation

Simplify the above equation to convert it to the slope-intercept form \(y=mx+b\). Distribute \(3\) in the right side of the equation to yield \(y - 3 = 3x - 3\), so the equation simplifies to \(y = 3x\).

Sketch the line

Having established the equation, the line can be sketched. The line passes through the point \((1,3)\) and has a positive slope of \(3\), meaning it rises to the right. Additionally, since the y-intercept (b) is not present in the equation, indicating it's \(0\), the line passes through the origin \((0,0)\) and goes up through the given point.

Key concepts

Slope-intercept form

The slope-intercept form is one of the most common ways to express the equation of a line. This form is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis. This form is especially useful because it allows us to quickly identify both the slope and the y-intercept just from looking at the equation.
Having the equation in slope-intercept form makes graphing easier as well. You start by placing a point at \(b\) on the y-axis. Then, using the slope \(m\), which is the change in \(y\) over the change in \(x\), you can determine another point on the line by "rising" and "running" from the y-intercept:

• If the slope is positive, the line rises to the right.
• If the slope is negative, the line falls to the right.

Point-slope form

The point-slope form is another useful way to express the equation of a line, particularly when you know one point on the line and the slope. It is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a given point on the line.
This form is especially helpful when we're given a point and a slope, as in the exercise. You simply substitute the point's coordinates and the slope into the formula. This will give you an equation of the line from which you can quickly derive the slope-intercept form by simplifying if needed.
One advantage of point-slope form is that it directly incorporates the slope, clearly showing how changes in \(x\) affect \(y\) and indicating the direction of the line based on the given point.

Graphing linear equations

Graphing linear equations involves plotting the equation on a coordinate plane to visualize the line it represents. When graphing a line using the slope-intercept form, \(y = mx + b\), you start by plotting the y-intercept \(b\) on the y-axis.
Next, you use the slope \(m\). Interpreted as a ratio \(\frac{rise}{run}\), it tells you how much the line "rises" or "falls" vertically for each "run" horizontally. For example:

• A slope of \(3\) means move up \(3\) units for every \(1\) unit you move to the right.
• If the slope were \(-3\), you would move down \(3\) units for each unit you move to the right.

Once the slope is applied from the y-intercept, you can mark another point on the graph and draw the line through both points. The line extends infinitely in both directions.

Slope of a line

The slope of a line is a measure of its steepness and direction. Slope is crucial in understanding and describing a line's behavior. It is calculated as the ratio between the vertical change (rise) and horizontal change (run) between any two points on the line.
Mathematically, slope \(m\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Positive slope values indicate that the line ascends from left to right, while negative values show it descends. A slope of zero means the line is completely horizontal, indicating no rise as \(x\) changes, whereas an undefined slope suggests a vertical line.
In the given exercise, the slope provided is \(3\). This positive slope indicates a line that rises steeply to the right, showing the consistent increase in \(y\) as \(x\) increases. Understanding the slope helps in visualizing how the line appears on a graph and predicting the line’s direction.