Download Beginning Algebra

Linear Equation in Two Variables
A linear equation in two variables is an equation that
can be written in the form
Ax + By = C
where A, B, and C are real numbers and A and B not
both 0. The graph of a linear equation in two
variables is a straight line.
The form Ax + By = C is called standard form.
Martin-Gay, Beginning Algebra, 5ed
2
Martin-Gay, Beginning Algebra, 5ed
3
Example
Graph the linear equation 2x – y = – 4.
Let x = 1.
2(1) – y = – 4
2–y=–4
–y=–4–2=–6
y=6
Replace x with 1.
Simplify the left side.
Subtract 2 from both sides.
Multiply both sides by – 1.
One solution to the equation is (1, 6).
Martin-Gay, Beginning Algebra, 5ed
4
Example continued:
Graph the linear equation 2x – y = – 4.
For the second solution, let y = 4.
2x – 4 = – 4
2x = – 4 + 4
2x = 0
x=0
Replace y with 4.
Add 4 to both sides.
Simplify the right side.
Divide both sides by 2.
The second solution is (0, 4).
Martin-Gay, Beginning Algebra, 5ed
5
Example continued:
Graph the linear equation 2x – y = – 4.
For the third solution, let x = – 3.
2(– 3) – y = – 4
–6–y=–4
–y=–4+6=2
y=–2
Replace x with – 3.
Simplify the left side.
Add 6 to both sides.
Multiply both sides by – 1.
The third solution is (– 3, – 2).
Martin-Gay, Beginning Algebra, 5ed
6
y
Example continued:
(1, 6)
(0, 4)
Now plot all three of the
solutions (1, 6), (0, 4) and
(– 3, – 2).
x
Draw the line with
arrows that contains
the three points.
(– 3, – 2)
Martin-Gay, Beginning Algebra, 5ed
7
Martin-Gay, Beginning Algebra, 5ed
8
Martin-Gay, Beginning Algebra, 5ed
9
Example Graph y = 2
Solution
Writing in slope-intercept form: 0 • x + y = 2. No matter what
number we choose for x, we find that y must equal 2.
y=2
Choose any number for x
x
y
(x, y)
0
4
2
2
2
(0, 2)
(4, 2)
4
(4 , 2)
y must always be 2
Martin-Gay, Beginning Algebra, 5ed
10
Graph y = 2
When we plot the ordered
pairs (0, 2), (4, 2) and
(4, 2) and connect the points,
we obtain a horizontal line.
Any ordered pair of the form
(x, 2) is a solution, so the line
is parallel to the x-axis.
Martin-Gay, Beginning Algebra, 5ed
11
Example Graph x = 2
Solution
We regard the equation x = 2 as x + 0 • y = 2. We make up a
table with all 2 in the x-column.
x = 2
x must be 2
x
2
2
2
y
4
1
4
(x, y)
(2, 4)
(2, 1)
(2, 4)
Any number can be used for y
Martin-Gay, Beginning Algebra, 5ed
12
Graph x = 2
When we plot the ordered
pairs (2, 4), (2, 1), and
(2, 4) and connect them,
we obtain a vertical line.
Any ordered pair of the form
(2, y) is a solution. The line
is parallel to the y-axis.
Martin-Gay, Beginning Algebra, 5ed
13
Martin-Gay, Beginning Algebra, 5ed
14