Characteristics of Straight Line Graphs
Equations with two unknowns can be plotted in a two-dimensional graphing system. When an
equation with two unknowns (variables), both to the first degree (where variables have an
exponent of one), is plotted on a rectangular coordinate system, it graphs as a straight line. We
call this kind of equation a linear equation. Some examples of linear equations are:
x + 2y = 8;
x = 3;
x–y=2
y = -4;
Each equation represents a specific relationship between the variables x and y. If we find values
of x and y that satisfy the equation, we can plot the pairs of values on a two dimensional graph.
Let’s find some values of x and y that make –x + y = 2 a true statement.
X
Y
-x + y = 2
0
1
2
3
4
5
2
3
4
5
6
7
0+2=2
-1 + 3 = 2
-2 + 4 = 2
-3 + 5 = 2
-4 + 6 = 2
-5 + 7 = 2
Assuming that each pair of values represents the coordinates of a point we now have the points
(0, 2), (1, 3), (2, 4), (3, 5), (4, 6), (5, 7)
Plotting these points on a rectangular coordinate system we have:
Graph of -x + y = 2
8
7
y axis
6
5
4
3
2
1
0
0
1
2
3
4
5
6
x axis
A straight line may be extended in either direction to infinity. This means that there are an infinite
number of combinations of x and y that will satisfy the equation. Can you find two more
combinations?
Slope
Notice that the line has pitch to it. If this was the roof line of a house we might talk about this
pitch as rise over run, or a ratio of the vertical distance (rise) divided by the horizontal distance
(run) as we move from point to point along the line. We can calculate what we refer to as slope
by setting up a ratio of the difference between the y coordinates divided by the difference
between the x coordinates for two points:
Slope = rise = y2 – y1
run
x2 – x1
Using the first two point in our table (0, 2) and (1, 3) for (x1, y1) and (x2, y2) to calculate slope
gives us the following:
Slope = 3 – 2 =
1–0
1 = 1
1
So, this line has a slope of positive one. A line with a positive slope slants up and to the right.
Slope is an important concept in determining trends. Another way to calculate slope is to write
the equation in terms of y giving us the standard form: y = mx + b
When we write the equation for a line in this form, the coefficient m is the slope and the constant
b is the y-intercept (the point where the line crosses the y axis).
Solving –x + y = 2 for y involves the following steps:
1. Add x to both sides of the equation:
y = x + 2 OR y = 1x + 2
2. The equation is now in the form y = mx + b. The coefficient of x is 1 which is the slope, and
the y-intercept is 2. If you look at the graph of the equation you can see that the line crosses the y
axis at 2.
Exercises:
In the following linear equations, identify the slope and the y intercept. Draw the lines on a graph
using this information
y = 2x + 3
y=x+4
y = 3x + 1