A linear equation is an equation whose solutions form a straight line on the coordinate plane.
(Linear = Line)

Linear equation: y = 2x + 3

Solutions as seen in a table:

These x and y values make the equation true.

x- value
y-value
ordered pair
(x, y)
-3
-3
(-3, -3)
-2
-1
(-2, -1)
-1
1
(-1, 1)
0
3
(0, 3)
1
5
(1, 5)
2
7
(2, 7)
3
9
(3, 9)


When the ordered pairs are plotted on a graph (the coordinate plane), a straight line will be formed.
drawline
Any point on this line is a solution
to the equation y = 2x + 3.

We can see from the example above, that our linear equation can be viewed three ways:
as an equation, as a list of ordered pairs in a table, and as a graph.

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Let's look at an application problem.

Mr. Carpenter is driving his car down a highway at a constant speed of 65 mph.

The equation to determine the distance he has traveled is d = 65t
where d = distance traveled in miles
and t = the time traveled in hours.

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The Table of Ordered Pairs:

t - value
(hours)
d-value
(miles)
ordered pair
(t, d)
0
0
(0,0)
1
65
(1,65)
2
130
(2,130)
3
195
(3,195)
4
260
(4, 260)
8
520
(8, 520)
10
650
(10, 650)

The Graph of Distance TraveledcarG

• The graph for this problem only shows the first quadrant, where the values of time and distance are positive. In real world problems, time and distance are positive quantities.

• Any ordered pair on the graphed line will be a solution to the equation d = 65t.

• Note that the scale on the t-axis is one unit and the scale on the d-axis is 65 units.

• Note that for every hour traveled, the distance traveled increases by 65 miles.

• Since the distance traveled is dependent upon the amount of time Mr. Carpenter drives, d is the dependent variable in this problem and t is the independent variable.



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