Math in Our World
Section 7.1
The Rectangular Coordinate
System and Linear Equations in
Two Variables
Learning Objectives
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Plot points in a rectangular coordinate system.
Graph linear equations.
Find the intercepts of a linear equation.
Find the slope of a line.
Graph linear equations in slope-intercept form.
Graph horizontal and vertical lines.
Find linear equations that describe real-world
situations.
Rectangular Coordinate System
The foundation of graphing in mathematics is a system for
locating data points using a pair of perpendicular number
lines. We call each one an axis.
The horizontal line is called the x
axis, and the vertical line is called
the y axis. The point where the
two intersect is called the origin.
Collectively, they form what is
known as a rectangular coordinate
system, sometimes called the
Cartesian plane.
The two axes divide the plane into four regions called quadrants.
They are numbered using Roman numerals I, II, III, and IV.
Rectangular Coordinate System
The location of each point is given by a pair of numbers
called the coordinates, and are written as (x, y), where
the first number describes a number on the x-axis and the
second describes a number on the y-axis.
The coordinates of the origin
are (0, 0).
A point P whose x coordinate
is 2 and whose y coordinate is
5 is written as P = (2, 5). It is
plotted by starting at the origin
and moving two units right and
five units up.
EXAMPLE 1
Plotting Points
Plot the points (5, –3), (0, 4), (–3, –2), (–2, 0), and
(2, 6).
SOLUTION
To plot each point, start
at the origin and move
left or right according to
the x value, and then up
or down according to the
y value.
Identifying Coordinates
Given a point on the
plane, its coordinates
can be found by
drawing a vertical line
back to the x axis and a
horizontal line back to
the y axis. For example,
the coordinates of point
C shown are (–3, 4).
EXAMPLE 2
Finding the Coordinates of
Points
Find the coordinates of each point shown on the
plane.
SOLUTION
A = (1, 4)
B = (–2, 6)
C = (–5, 0)
D = (0, 3)
E = (–4, –4)
Linear Equations in Two Variables
An equation of the form ax + by = c, where a,
b, and c are real numbers, is called a linear
equation in two variables.
Let’s look at the example 3x + y = 6. If we choose a
pair of numbers to substitute in for x and y, the
resulting equation will be either true or false.
For example, for x = 1 and y = 3, the equation
becomes 3(1) + 3 = 6, which is a true statement.
We call the pair (1, 3) a solution to the equation,
and say the pair satisfies the equation.
Linear Equations in Two Variables
Listed below are a handful of pairs that satisfy the
equation 3x + y = 6.
(0, 6)
(1, 3)
(2, 0)
(3, – 3)
If we plot the points corresponding to
these pairs, all of the points appear
to line up in a straight line pattern. If
we connect these points with a line,
the result is called the graph of the
equation. The graph is a geometric
representation of every pair of
numbers that is a solution to the
equation.
(4, – 6)
EXAMPLE 3
Graphing a Linear Equation in
Two Variables
Graph x + 2y = 5
SOLUTION
Only two points are necessary to find the graph of a line,
but it’s a good idea to find three.
To find pairs of numbers that make the equation true, we
will choose some numbers to substitute in for x, then solve
the resulting equation to find the associated y.
EXAMPLE 3
Graphing a Linear Equation in
Two Variables
SOLUTION
In this case, we chose x = –1, x = 1,
and x = 5, but any three will do.
Three points on the graph are (1, 3), (1, 2), and (5, 0). We
plot those three points and draw a straight line through them.
Intercepts
The point where a graph crosses the x axis is
called the x intercept. The point where a graph
crosses the y axis is called the y intercept. Every
point on the x axis has y coordinate zero, and every
point on the y axis has x coordinate zero, so we get
the following rules.
Finding Intercepts
To find the x intercept, substitute 0 for y and solve
the equation for x.
To find the y intercept, substitute 0 for x and solve
the equation for y.
EXAMPLE 4
Finding Intercepts
Find the intercepts of 2x – 3y = 6, and use them to
draw the graph.
SOLUTION
To find the x intercept, let y = 0
and solve for x.
The x intercept has the
coordinates (3, 0).
To find the y intercept, let x = 0
and solve for y.
The y intercept has the
coordinates (0, – 2).
EXAMPLE 4
Finding Intercepts
SOLUTION
Now we plot the points (3, 0)
and (0, – 2), and draw a
straight line through them.
(It would still be a good idea
to find one additional point to
check your work. If the three
points don’t line up, there
must be a mistake.)
Slope
The slope of a line (designated by m) is
y 2  y1
m
x2  x1
where (x1, y1) and (x2, y2) are two points on the line.
The “slope” can be defined as the “rise” (vertical
height) divided by the “run” (horizontal distance) or
as the change in y with respect to the change in x.
EXAMPLE 5
Finding the Slope of a Line
Find the slope of a line passing through the points
(2, 3) and (5, 8).
SOLUTION
Designate the points as follows
Substitute into the formula
That means the line is rising 5 feet vertically for every 3 feet
horizontally.
Slope
When finding slope, it doesn’t matter which of the
two points you choose to call (x1, y1) and which you
call (x2, y2). But the order of the subtraction in the
numerator and denominator has to be consistent
If the line goes “uphill” from left to right, the slope will be
positive. If a line goes “downhill” from left to right, the slope
will be negative. The slope of a vertical line is undefined.
The slope of a horizontal line is 0.
EXAMPLE 6
Finding Slope Given the
Equation of a Line
Find the slope of the line 5x – 3y = 15.
SOLUTION
Find the coordinates of any two points on the line. In this
case, we choose the intercepts, which are (3, 0) and (0, – 5).
Then substitute into the slope formula.
Slope-Intercept Form
The slope-intercept form for an equation in two
variables is y = mx + b, where m is the slope and
(0, b) is the y intercept.
If we start with the equation 5x – 3y = 15 from Example 6
and solve the equation for y,
Notice that the coefficient of x is 5/3, which is the same as
the slope of the line, as found in Example 6.
EXAMPLE 7
Using Slope-Intercept Form to
Draw a Graph
Graph the line
SOLUTION
The slope is 5/3 and the y intercept is
(0, –6). Starting at the point (0, –6),
we move vertically upward 5 units
for the rise, and move horizontally 3
units right for the run. That gives us
second point (3, –1). Then draw a
line through these points.
To check, notice that (3, –1) satisfies
the equation.
Horizontal and Vertical Lines
Think about what the equation y = 3 says in words:
that the y coordinate is always 3. This is a line
whose height is always 3, which is a horizontal line.
Similarly, an equation like x = – 6 is a vertical line
with every point having x coordinate – 6.
EXAMPLE 8
Graphing Vertical and
Horizontal Lines
Graph each line: (a) x = 5 and (b) y = – 3.
SOLUTION
(a) The graph of x = 5 is a
vertical line with every point
having x coordinate 5. We draw
it so that it passes through 5 on
the x axis.
(b) The graph of y = – 3 is a
horizontal line with every point
having y coordinate 3. We draw
it so that it passes through – 3 on
the y axis.
EXAMPLE 9
Finding a Linear Equation
Describing Cab Fare
The standard fare for a taxi in one city is $5.50,
plus $0.30 per mile. Write a linear equation that
describes the cost of a cab ride in terms of the
length of the ride in miles. Then use your equation
to find the cost of a 6-mile ride, an 8.5-mile ride,
and a 12-mile ride.
EXAMPLE 9
Finding a Linear Equation
Describing Cab Fare
SOLUTION
The first quantity that varies in this situation is the length of
the trip, so we will assign variable x to number of miles. The
corresponding quantity that changes is the cost, so we will
let y = the cost of the ride. Since each mile costs $0.30, the
total mileage cost is 0.30x. Adding the upfront cost of $5.50,
the total cost is given by
y = 0.30x + 5.50.
Now let’s evaluate for x = 6, x = 8.5 and x = 12 miles,
Slope and Rate of Change
The slope of any line tells us the rate at
which y changes with respect to x.
EXAMPLE 10
Finding a Linear Equation
Describing Distance
After a brisk bike ride, you take a break and set out
for home. Let’s say you start out 15 miles from
home and decide to relax on the way home and
ride at 9 miles per hour. Write a linear equation
that describes your distance from home in terms of
hours, and use it to find how long it will take you to
reach home.
EXAMPLE 10
Finding a Linear Equation
Describing Distance
SOLUTION
In this case, we know two key pieces of information: at time
zero (when you start out for home) the distance is 15, and
the rate at which that distance is changing is – 9 miles per
hour (negative because the distance is decreasing). The
rate is the slope of a line describing distance, and the
distance when time is zero is the y intercept.
Let y = distance and x = hours after starting for home.
y = – 9x + 15.
EXAMPLE 10
Finding a Linear Equation
Describing Distance
SOLUTION
You reach home when the distance (y) is zero, so
substitute in y = 0 and solve for x:
It will take 1 hour and 40 minutes to get home.