Algebra – Part 8: Graphing Linear Equations

A linear equation represents the equation of a line, and graphing a linear equation means graphing a line. To graph a line, one must first put the equation into either point-slope form or slope-intercept form.

Point-slope form:

y − y1 = m(x − x1)

Slope-intercept form:

y = mx + b

It is important to recall that the symbols “m” and “b” are used in linear equations. “m” represents the slope of the line, while “b” represents the y-intercept, which is the y-coordinate of the point where the line intersects the y-axis.

When graphing linear equations, the intercepts of the line can be used as a helpful tool. The y-intercept is represented by “b” and the x-intercept can be found by setting y equal to zero. The intercepts of the line are the points where the line crosses the major axes.

Sample 1:

This equation is in slope-intercept form, but the y-intercept is missing.
However, we could actually rewrite the equation of the line like this:

When written in this manner, the value of both sides of the equation remains unchanged. However, it becomes apparent that the y-intercept is equal to zero, indicating that the line passes through the origin.

Sample 2:

The equation is already in slope-intercept form, making it easy to graph. The y-intercept “b = -2” can be plotted by placing a point at (0, -2), which is two units below the origin on the y-axis.

The slope of the line is “m = 3”, which means that for every unit to the right, the line moves up three units. We can plot a new point by moving up 3 units and to the right 1 unit.

So a sketch of the line is:

Sample 3:

We are going to look at a line on a graph. The line touches the y-axis at a spot where y is equal to 2. To figure out how steep the line is, we pick another point on the line, like (3, -3). To get from the spot where y is 2 to the point (3, -3), we go down 5 steps and then to the right 3 steps. So the steepness of the line is -5/3.

The equation of the line is:

Sample 4:

If the slope of the line is m = − 3/4 and it passes through the point (x1, y1) = (1, − 2), how can we use the slope to find another point on the graph?

One way to find another point on the line is to move 3 units up and 4 units to the left, which would place the second point at (1 – 4, -2 + 3) = (-3, 1). Alternatively, we can move 3 units down and 4 units to the right, which would place the second point at (1 + 4, -2 – 3) = (5, -5).

Additionally, we can continue this process and plot more points along the line by repeatedly moving up and to the left or down and to the right.

Final challenge:

The equation x + y = 2 is a simple linear equation in two variables, and it represents a straight line in the x-y plane. To graph this equation, you can follow the step:

Plot the y-intercept: Start by finding the y-intercept of the line. This is the point where the line crosses the y-axis, and it can be found by setting x = 0. In this case, the equation becomes y = 2 – x, and when x = 0, y = 2. So the y-intercept is at the point (0, 2).

So, the answer is A.