Question

In Problems 25-32, graph the line that contains the point P and has slope \(\mathrm{m}\). $$ P=(1,2) ; m=3 $$

Short answer

The equation of the line is \( y = 3x - 1 \).

Step-by-step solution

Understand the given information

You are provided with a point, P = (1, 2), and a slope, m = 3. These will help in generating the equation of the line.

Use the point-slope form of the line equation

The point-slope form is given by \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) = (1, 2) \) and \( m = 3 \).

Substitute the values into the point-slope form

Replace \( x_1 \) with 1, \( y_1 \) with 2, and \( m \) with 3: \[ y - 2 = 3(x - 1) \]

Simplify the equation to the slope-intercept form

Expand and solve for \( y \): \[ y - 2 = 3x - 3 \] Add 2 to both sides: \[ y = 3x - 1 \]

Graph the line

With the equation \( y = 3x - 1 \), plot the point \( (1, 2) \). Then, use the slope, which means for every 1 unit increase in \( x \), \( y \) increases by 3 units, to find another point, such as the point \( (2, 5) \). Draw a line through these points.

Key concepts

Point-Slope Form Explained

The point-slope form of a linear equation is a convenient way to write the equation of a line when you know one point on the line and the slope. The general formula for the point-slope form is: y - y_1 = m(x - x_1)
where x_1 and y_1 represent the coordinates of a known point on the line, and m is the slope of the line.
Why is it useful? It allows us to directly plug in the values we have without needing to rearrange the equation.
Let's break this down with our example: If Point P=(1,2) and Slope m=3 then our equation looks like: y - 2 = 3(x - 1). This setup makes it easier to graph or transform into other forms of the line equation.

Understanding Slope-Intercept Form

The slope-intercept form of a line equation is possibly the most famous form. Its general structure is: y = mx + b
where: m is the slope of the line, b is the y-intercept (the point where the line crosses the y-axis).
Converting from the point-slope form to the slope-intercept form involves a bit of simplification. Take the example from our previous section: Start with: y - 2 = 3(x - 1)
Step-by-step to simplify we get: y - 2 = 3x - 3
Add 2 to both sides: y = 3x - 1. Now you have the equation in slope-intercept form. Why is it helpful? The form y = mx + b is really simple for graphing. You can easily see the slope and where the line crosses the y-axis.

Plotting on the Coordinate Plane

To graph a line, understanding the coordinate plane is crucial. The coordinate plane is a two-dimensional surface formed by: x-axis (horizontal line), y-axis (vertical line).
Every point on the plane is represented as (x, y). To graph the equation y = 3x - 1, follow these steps:

• Identify the y-intercept (the point where the line crosses the y-axis, which is -1 in this example). Plot this point at (0, -1).
• Use the slope to find another point. Slope m=3 means for every 1 unit increase in x, y increases by 3. From the point (0, -1), move 1 unit right (to x=1) and 3 units up (to y=2), giving you the point (1, 2).
• Draw the line through these points.

Always remember to use both the y-intercept and the slope to plot the correct points on the coordinate plane.