Download Linear Equations

Linear Equations
Identifying a Linear Equation
An equation is linear if it can be written in the
form Ax + By = C
● The exponent of each variable is 1.
● The variables are added or subtracted.
● A or B can equal zero but not both.
● A > 0 (A can’t be negative.)
● Besides x and y, other commonly used variables
are m and n, a and b, and r and s.
● There are no radicals in the equation.
● Every linear equation graphs as a line.
Examples of linear equations
Equation is in Ax + By =C form
2x + 4y =8
6y = 3 – x
Rewrite with both variables
on left side … x + 6y =3
x=1
B =0 … x + 0  y =1
-2a + b = 5
Multiply both sides of the
equation by -1 … 2a – b = -5
4x  y
 7
3
Multiply both sides of the
equation by 3 … 4x –y =-21
Examples of Nonlinear Equations
The following equations are NOT in the
standard form of Ax + By =C:
4x2 + y = 5
x4
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
Forms of linear equations.
Ax + By =C is the
standard form of a
linear equation. The
slope is –A/B, the
x-int. is C/A and
the y-int. is C/B.
(y – y1) = m(x - x1) is
point-slope form.
(equation is usually not left in this
form)
y= mx+b is the slopeintercept form of a
linear equation. The
slope is m and b is
the y-intercept.
Writing equations in either form.
To write an equation
in slope-intercept
form you simply
solve for y.
To write an equation
in standard form
you get both
variable terms on
the left and the
constant on the
right. Remember
that you can’t have
fractions and the
leading coefficient
must be positive.
x and y -intercepts
●
The x-intercept is the point where a line crosses
the x-axis.
The general form of the x-intercept is (x, 0).
The y-coordinate will always be zero.
●
The y-intercept is the point where a line crosses
the y-axis.
The general form of the y-intercept is (0, y).
The x-coordinate will always be zero.
Finding the x-intercept
●
●
●
●
Find the x-intercept for the equation 2x + y = 6
We know that at the x-intercept (where the graph
touches the x-axis) the value of y is zero.
Plug in 0 for y and simplify.
2x + 0 = 6
2x = 6
x=3
So (3, 0) is the x-intercept of the line.
Finding the y-intercept
●
●
●
●
Find the y-intercept for the equation 2x + y = 6
We know that at the y-intercept (where the graph
touches the y-axis) the value of x is zero.
Plug in 0 for x and simplify.
2(0) + y = 6
0+y=6
y=6
So (0, 6) is the y-intercept of the line.
To summarize….
●
To find the x-intercept, plug in 0
for y.
●
To find the y-intercept, plug in 0
for x.
Find the x and y- intercepts
of x = 4y – 5
●
●
●
x-intercept:
Let y = 0
x = 4y - 5
x = 4(0) - 5
x=0-5
x = -5
(-5, 0) is the
x-intercept
●
●
y-intercept:
Let x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
5
=y
4
●
5
(0, 4 )
is the
y-intercept
Find the x and y-intercepts
of g(x) = -3x – 1*
●
●
●
x-intercept
Let y = 0
g(x) = -3x - 1
0 = -3x - 1
1 = -3x
1

=x
3
1
(  3 , 0) is the
x-intercept
*g(x) is the same as y
●
●
●
y-intercept
Let x = 0
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
(0, -1) is the
y-intercept
Find the x and y-intercepts of
6x - 3y =-18
●
●
●
x-intercept
Plug in y = 0
6x - 3y = -18
6x -3(0) = -18
6x - 0 = -18
6x = -18
x = -3
(-3, 0) is the
x-intercept
●
●
●
y-intercept
Plug in x = 0
6x -3y = -18
6(0) -3y = -18
0 - 3y = -18
-3y = -18
y=6
(0, 6) is the
y-intercept
Find the x and y-intercepts
of x = 3
●
x-intercept
●
●
Plug in y = 0.
●A
There is no y.
x = 3 is a vertical line
so x always equals 3.
●
●
y-intercept
vertical line never
crosses the y-axis.
●
There is no y-intercept.
(3, 0) is the x-intercept.
x
Find the x and y-intercepts
of y = -2
●
x-intercept
Plug in y = 0.
y cannot = 0 because
y = -2.
● y = -2 is a horizontal
line so it never crosses
the x-axis.
●
●There
●
y-intercept
●
y = -2 is a horizontal line
so y always equals -2.
●
(0,-2) is the y-intercept.
x
is no x-intercept.
y
Shortcut for finding the intercepts
If the equation is in standard form
ax + by = c
you can find the x- intercept by dividing
the constant, c, by the x coefficient, a.

c
x- intercept =
a
Shortcut for finding the intercepts
If the equation is in standard form
ax + by = c
you can find the y- intercept by dividing
the constant, c, by the y coefficient, b.

c
y- intercept =
b
Slope
Slope is the steepness of a line.
Given a line in slope-intercept form,
y = mx + b, the slope is m.
Given a line in standard form, ax+by = c ,
a
the slope is  b (or you can put the equation
into slope-intercept form to find the slope)