# Lines and Linear Equations

By a linear equation we mean an equation of the form

y = ax + b (1)

where
.
The distingushing feature is the single power of the variable x . A linear

equation represents a line, that is the equation determines points in the plane
which we

can connect with a straight line. Moreover, given the graph of a line we can
write down its

(linear) equation. This requires two ingredients: the slope of the line and its
y-intercept.

The slope represents the change along the line along the y-axis versus the
change along

the x-axis. Given two points in in the plane (x_{1}, y_{1}) and (x_{2}, y_{2}) the slope of
the line

through them is found by computing

(2)

**Example** Find the slope of the line passing through the points (0, 1) and (−1,
1).

**Let **(x_{1}, y_{1}) = (−1, 1) **and let** (x_{2}, y_{2}) = (0, 1). **Then**

Therefore, the slope of the line through the points (0, 1) and (−1, 1) is m = 0.

The letter m is commonly used for the slope of a line, thus equation (1) becomes

y = mx + b (3)

Whenever a linear equation has the above form we say that is is in the
slope- intercept form .

Warning It is important to designate a start- and an end-point in slope
computations. In

the above example the same answer is obtained when (x_{1}, y_{1}) = (0, 1) and let
(x_{2}, y_{2}) =

(−1, 1). A common mistake is to take the points out of order .

Lines parallel to each other have the same slope, while lines perpendicular to
each

other have slopes which multiply to −1. That is, if m1 and m2 are the slopes of
two

perpendicular lines, then m1 ·m2 = −1. In other words, to find the slope of a
line perpendicular

to a given line one needs to find the negative reciprocal of the given slope. We
will

use the superscript ? to denote perpendicular slopes.

**
Example** Find the slope of a line perpendicular to the line given by y = 3x + 1.

From the slope-intercept form of the line we read off the slope. We have m = 3 and

.

Horizontal lines have a zero slope . These line have the form y = c for some real

number c. The line in the first example is a horizontal line with equation y = 1. Vertical

lines are of the from x = k, where k is any real number . Vertical lines are perpendicular to

horizontal lines, however, their slopes do not multiply to −1. This is because vertical lines

have undefined slope: on a vertical line the change along the x-axis between consecutive

points is 0, thus in the process of computing the slope of a vertical line we would be

dividing by 0.

**Example**Find the equation of a line perpendicular to the line in the first example.

We already know that the equation of the line is y = 1. From this equation we read off

the slope (the coefficient of x ) to be 0. We cannot take the negative reciprocal without

dividing by zero, but any vertical line will be perpendicular to y = 1. We can choose

x = 1. The lines y = 1 and x = 1 are perpendicular and intersect at the point (1, 1).

The final ingredient in determining the equation of a line is the y-intercept. This is

the point where the graph of the line intersects the y-axis (the line x = 0) and is obtained

by letting x = 0 in the line’s equation. In the above example x is identically 0, thus the

y-intercept is b = 1. In general, y-intercepts have the form (0, p). Similarly, the x-intercept

is the point on the graph of the line which intersects the x-axis (the line y = 0) and is

obtained by letting y = 0 in the line’s equation. x-intercepts have the form (q, 0).

**Example**Find the equation of a line parallel to the line 5x + y = 7 and passing through the

origin.

Writing the equation in slope-intercept form we have y = −5x + 7. We read off the slope

to be m = −5. A parallel line will have the same slope and since the parallel line must

pass through (0, 0) the y-intercept is b = 0. Therefore, the desired equation is y = −5x.

It is possible to write down the equation of a line without explicitly calculating its

y-intercept (see exercise 7). The point-slope form of a line through a point (x

_{1}, y

_{1}) is

y − y

_{1}= m(x − x

_{1}) (4)

Here the y-intercept is disguised as b = y

_{1}− mx

_{1}.

**Systems of Linear Equations**

Given two or more linear equations we call the point (if one exists) where all the lines

intersect the solution to this system of linear equations. In a previous example we saw

that the lines x = 1 and y = 1 intersect at (1, 1). In other words, the solution to the system

of equations

is the point (1, 1).

**Example**The y-intercept of the line y = mx+b is the solution to the system of equations

given by

and the x-intercept is the solution to

**Example**The solution to the system of equations

is the point (18,−11).

The main methods of solving a system of linear equations are elimination and substitution .

When it is easy enough to solve for one variable , as in the example with the

x-intercept, we do so and then make a substitution in the other equation, thereby obtaining

an easy to solve linear equation in one variable. Remember that multiplying both

sides of an equation by a number does not change the equation. To solve the system

by elimination we would first multiply the bottom equation by -6 and obtain the equivalent

system

Adding both equations eliminates the variable y and we have 2x = 36 whence x = 18. It

then follows that y = −11 and therefore the solution to the system is (18,−11).

**Note**It is absolutely vital to check you answer by making sure that it is the solution to

every equation.

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