Missing Coordinates and Graphing with Slope | Grade 8 Math | LevelKite

You already know how to find the slope of a line from two points: divide the change in $y$ (the rise) by the change in $x$ (the run). A slope of $2$ means that each time $x$ goes up by $1$ , $y$ goes up by $2$ . So if $x$ goes up by $3$ , $y$ goes up by $2 \times 3 = 6$ .

In this lesson you use slope the other way around. When you already know the slope, it can tell you something that is missing:

a missing coordinate of a point on the line
the graph of the whole line, starting from just one point

Both jobs rest on one fact. On a straight line that is not vertical, the slope is the same between any two points. So the slope tells you exactly how much $y$ must change whenever $x$ changes.

Key concept

Slope links the change in y to the change in x

Take two different points $(x_1, y_1)$ and $(x_2, y_2)$ on a line that is not vertical. The slope $m$ is the change in $y$ divided by the change in $x$ . The change in $x$ cannot be $0$ , because division by zero is undefined. That is why a vertical line has an undefined slope, and why this method does not apply to it.

Because $x_2 - x_1$ is not $0$ , you may multiply both sides of the slope equation by it. In words, the change in $y$ equals the slope times the change in $x$ .

Know the slope and the change in $x$ , and you can find the change in $y$ . Know the slope and the change in $y$ , and you can divide by the slope to find the change in $x$ , as long as the slope is not $0$ .

m = \frac{y_2 - y_1}{x_2 - x_1}, \quad \text{so} \quad y_2 - y_1 = m\,(x_2 - x_1) \quad \text{when } x_2 \neq x_1

Worked example

Find a missing y-coordinate

A line has slope $2$ . It passes through the points $(1, 3)$ and $(4, y)$ . Find the value of $y$ .

Find the change in x from the first point to the second. It is 3, which is not 0, so the two points are not on a vertical line and the slope is defined.
$4 - 1 = 3$
Write the definition of slope with the values you know: the change in y is y - 3, the change in x is 3, and the slope is 2.
$\frac{y - 3}{3} = 2$
Multiply both sides by 3 to undo the division. Multiplying both sides by the same nonzero number keeps the equation true. This is the key idea at work: the change in y equals the slope times the change in x.
$y - 3 = 2 \times 3 = 6$
Add 3 to both sides so that y is alone. Adding the same number to both sides keeps the equation true.
$y = 6 + 3 = 9$
Check by finding the slope between (1, 3) and (4, 9). It equals the given slope, 2.
$\frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$

Answer

The missing coordinate is $y = 9$ , so the second point is $(4, 9)$ .

Line A has slope

2

and passes through

(0, 1)

,

(1, 3)

and

(4, 9)

.

The figure above shows the line from the first worked example as Line A. The horizontal axis shows $x$ -values, the vertical axis shows $y$ -values, and each grid step is $1$ unit. Line A passes through the given point $(1, 3)$ and the point you found, $(4, 9)$ . Moving from $(1, 3)$ to $(4, 9)$ takes you $3$ units right and $6$ units up, and $6 \div 3 = 2$ , the slope.

Decide whether a point is on a line by calculating with its exact coordinates, not by judging how the picture looks. For example, step back from $(1, 3)$ by $1$ unit in $x$ . Then $y$ must drop by $2 \times 1 = 2$ , which gives the third marked point, $(0, 1)$ . So Line A crosses the $y$ -axis at $(0, 1)$ , not at the origin, the point $(0, 0)$ .

Worked example

Find a missing x-coordinate

A line has slope $-\frac{3}{2}$ . It passes through the points $(-2, 5)$ and $(x, -1)$ . Find the value of $x$ .

Find the change in y from the first point to the second. Use the order second point minus first point, and keep that order for x too.
$-1 - 5 = -6$
Write the change in x in the same order. It contains the unknown. The slope is a number, -3/2, so the line is not vertical and this change in x is not 0.
$x - (-2) = x + 2$
Use the key idea: the change in y equals the slope times the change in x. Put in the two changes and the slope.
$-6 = -\frac{3}{2}\,(x + 2)$
Divide both sides by the slope, -3/2. The slope is not 0, so this division is defined. Dividing by -3/2 is the same as multiplying by its reciprocal, -2/3, which is the same fraction turned upside down. A negative times a negative is positive.
$x + 2 = -6 \div \left(-\frac{3}{2}\right) = -6 \times \left(-\frac{2}{3}\right) = \frac{12}{3} = 4$
Subtract 2 from both sides so that x is alone.
$x = 4 - 2 = 2$
Check by finding the slope between (-2, 5) and (2, -1). The fraction -6/4 simplifies to -3/2, the given slope.
$\frac{-1 - 5}{2 - (-2)} = \frac{-6}{4} = -\frac{3}{2}$

Answer

The missing coordinate is $x = 2$ , so the second point is $(2, -1)$ .

Now use the same idea to graph a line, that is, to draw it accurately on a coordinate plane. Two different points fix exactly one straight line. So if you know one point and the slope, let the slope lead you to a second point.

Read the slope as a move: rise over run. A slope of $\frac{3}{4}$ says that a run of $4$ units to the right goes with a rise of $3$ units up. Start on the line, make that move, and you land on the line again. A whole-number slope such as $2$ is $\frac{2}{1}$ : run $1$ , rise $2$ .

Plot the point you know. Start there, even when it is not the origin.
Move by the run, then by the rise, and plot the point where you land.
Repeat the move to get a third point as a check. Then draw the straight line through the points.

This works for any line that is not vertical. A vertical line has an undefined slope, so it gives you no rise-over-run move to follow.

Worked example

Graph a line from one point and its slope

Graph the line that passes through $(-2, 5)$ and has slope $-\frac{3}{2}$ .

Plot the point you know, (-2, 5). It is 2 units left of the vertical axis and 5 units above the horizontal axis. This is where you start.
Read the slope as rise over run. Give the negative sign to the rise: a run of 2 goes with a rise of -3. So when x increases by 2, y decreases by 3. The run is not 0, so the slope is defined.
$-\frac{3}{2} = \frac{-3}{2}$
From (-2, 5), move 2 units right and 3 units down. That adds the run to the x-coordinate and the rise to the y-coordinate. Plot the point where you land.
$(-2 + 2,\; 5 + (-3)) = (0, 2)$
Make the same move again, this time from (0, 2), and plot the third point.
$(0 + 2,\; 2 + (-3)) = (2, -1)$
Step the other way to extend the line: 2 units left and 3 units up from (-2, 5). The slope is unchanged, because 3 divided by -2 also equals -3/2. Plot the point where you land.
$\frac{3}{-2} = -\frac{3}{2}, \qquad (-2 - 2,\; 5 + 3) = (-4, 8)$
Draw one straight line through the four points. Check it with the two points farthest apart, (-4, 8) and (2, -1). The slope between them is still -3/2.
$\frac{-1 - 8}{2 - (-4)} = \frac{-9}{6} = -\frac{3}{2}$

Answer

The graph is the straight line through $(-4, 8)$ , $(-2, 5)$ , $(0, 2)$ and $(2, -1)$ . It goes down $3$ units for every $2$ units it goes to the right.

Line B passes through

(-4, 8)

,

(-2, 5)

,

(0, 2)

and

(2, -1)

: each step is

2

units right and

3

units down.

The figure above shows the finished graph as Line B. Again the horizontal axis shows $x$ -values, the vertical axis shows $y$ -values, and each grid step is $1$ unit. The starting point is $(-2, 5)$ . Each move of $2$ units right and $3$ units down lands on the next marked point: first $(0, 2)$ , then $(2, -1)$ . One move the other way, $2$ units left and $3$ units up, lands on $(-4, 8)$ .

Look at $(2, -1)$ . It is the point you found with an equation in the second worked example, from the same slope and the same starting point. Finding a missing coordinate and graphing with slope are one idea used in two ways.

The negative sign goes with the rise or with the run, never both. Both $\frac{-3}{2}$ and $\frac{3}{-2}$ equal $-\frac{3}{2}$ , but $\frac{-3}{-2} = \frac{3}{2}$ , which is the slope of a different line.

Takeaway. On a line that is not vertical, the slope is the same between any two points, so the change in $y$ equals the slope times the change in $x$ . To find a missing coordinate, subtract to get the change you know, then multiply or divide by the slope to get the other change. Dividing needs a slope that is not $0$ . To graph a line, turn the slope into a rise-over-run move and repeat it from a point you know. Either way, check your result by recomputing the slope from exact coordinates.

In this lesson you relied on the slope staying the same all along a line. Next, you will use similar triangles to explain why slope is constant.