Linear Equations: Basics and Examples
08/15/2019
Agenda
Number Sense
Routine
Cornell Notes
Student/teacher
dialog
Number Sense Routine
5 + (−12)
10 + (−6)
−9 + (−13)
−12 + 17
Identifying a Linear Equation
Ax + By = C
●
The exponent of each variable is 1.
●
The variables are added or subtracted.
●
A
or
B
can equal zero.
●
A
> 0
●
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
line
ar equation graphs as a
line
.
Examples of linear equations
2
x
+ 4
y
=8
6
y
= 3 –
x
x
= 1
-2
a
+
b
= 5
Equation is in
A
x
+ B
y
=C
form
Rewrite with both variables
on left side …
x
+ 6
y
=3
B
=0 …
x
+ 0
y
=1
Multiply both sides of the
equation by -1 …
2
a
–
b
= -5
Multiply both sides of the
equation by 3 …
4
x
–
y
=-21
Examples of Nonlinear Equations
4x
2
+ y = 5
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
The following equations are NOT in the
standard form of
Ax
+
By
=
C
:
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
●
For the equation
2x + y = 6
, we know that
y
must equal 0. What must
x
equal?
●
Plug in 0 for
y
and simplify.
2
x
+ 0 = 6
2
x
= 6
x
= 3
●
So (3, 0) is the
x
-intercept of the line.
Finding the
y
-intercept
●
For the equation
2x + y = 6
, we know that
x
must equal 0. What must
y
equal?
●
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….
●
T
o
f
i
n
d
t
h
e
x
-
i
n
t
e
r
c
e
p
t
,
p
l
u
g
i
n
0
f
o
r
y
.
●
T
o
f
i
n
d
t
h
e
y
-
i
n
t
e
r
c
e
p
t
,
p
l
u
g
i
n
0
f
o
r
x
.
Find the
x
and
y-
intercepts
of
x
= 4
y
– 5
●
x
-intercept:
●
Plug in
y
= 0
x
= 4
y
- 5
x
= 4(0) - 5
x
= 0 - 5
x
= -5
●
(-5, 0) is the
x
-intercept
●
y
-intercept:
●
Plug in
x
= 0
x
= 4
y
- 5
0 = 4
y
- 5
5 = 4
y
=
y
●
(0, ) is the
y
-intercept
Find the
x
and
y
-intercepts
of g(
x
) = -3
x
– 1*
●
x
-intercept
●
Plug in
y
= 0
g(
x
) = -3
x
- 1
0 = -3
x
- 1
1 = -3
x
=
x
●
( , 0) is the
x
-intercept
●
y
-intercept
●
Plug in
x
= 0
g(x)
= -3(0) - 1
g(x
)
= 0 - 1
g(x)
= -1
●
(0, -1) is the
y
-intercept
*g(
x
)
is the same as
y
Find the
x
and
y
-intercepts of
6
x
- 3
y
=-18
●
x
-intercept
●
Plug in
y
= 0
6
x
- 3
y
= -18
6
x
-3(0) = -18
6
x
- 0 = -18
6
x
= -18
x
= -3
●
(-3, 0) is the
x
-intercept
●
y
-intercept
●
Plug in
x
= 0
6
x
-3
y
= -18
6(0) -3
y
= -18
0 - 3
y
= -18
-3
y
= -18
y
= 6
●
(0, 6) is the
y-intercept
Find the
x
and
y
-intercepts
of
x
= 3
●
y
-intercept
●
A vertical line never
crosses the
y
-axis.
●
There is no
y
-intercept.
●
x
-intercept
●
Plug in
y
= 0.
There is no y. Why?
●
x
= 3 is a
vertical
line
so
x
always equals 3.
●
(3, 0) is the
x
-intercept.
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 is no
x
-intercept.
●
y
-intercept
●
y
= -2 is a horizontal line
so
y
always equals -2.
●
(0,-2) is the
y
-intercept.
Graphing Equations
●
Example: Graph the equation -5
x
+
y
= 2
Solve for
y
first.
-5
x
+
y
= 2
Add 5
x
to both sides
y
= 5
x
+ 2
●
The equation
y
= 5
x
+ 2 is in slope-intercept form,
y = mx+b
. The
y
-intercept is 2 and the slope is 5.
Graph the line on the coordinate plane.
Graph
y
= 5
x
+ 2
Graphing Equations
Graph 4
x
- 3
y
= 12
●
Solve for
y
first
4
x
- 3
y
=12
Subtract 4
x
from both sides
-3y = -4
x
+ 12
Divide by -3
y
=
x
+
Simplify
y
=
x
– 4
●
The equation
y
=
x
- 4 is in slope-intercept form,
y=mx+b
. The
y
-intercept is -4 and the slope is .
Graph the line on the coordinate plane.
Graphing Equations
Graph
y
=
x
- 4
Graphing Equations
Reference
teachers.henrico.k12.va.us/math/hcpsal
gebra2/Documents/2-2/2006_2_2.ppt
This content covers the basics of linear equations, including their standard form Ax + By = C and how to identify them. It also discusses examples of linear and nonlinear equations, x and y-intercepts, and how to find them. The visual aids and explanations provided make it easier to grasp these fundamental concepts in mathematics.
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Presentation Transcript
08/15/2019 Agenda Number Sense Routine Cornell Notes Student/teacher dialog Number Sense Routine 5 + ( 12) 10 + ( 6) 9 + ( 13) 12 + 17
Identifying a Linear Equation Ax + By = C The exponent of each variable is 1. The variables are added or subtracted. A or B can equal zero. A > 0 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 2x + 4y =8 6y = 3 x x = 1 Equation is in Ax + By =C form Rewrite with both variables on left side x + 6y =3 B =0 x + 0 y =1 -2a + b = 5 Multiply both sides of the equation by -1 2a b = -5 4 x y = = Multiply both sides of the equation by 3 4x y =-21 7 3
Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: The exponent is 2 4x2 + y = 5 xy + x = 5 s/r + r = 3 There is a radical in the equation x = = 4 Variables are multiplied Variables are divided
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 For the equation 2x + y = 6, we know that y must equal 0. What must x equal? 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 For the equation 2x + y = 6, we know that x must equal 0. What must y equal? 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 y-intercept: x-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y = y 4 Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5 5 (-5, 0) is the x-intercept 5 4 (0, ) is the y-intercept
Find the x and y-intercepts of g(x) = -3x 1* y-intercept x-intercept Plug in x = 0 g(x) = -3(0) - 1 g(x)= 0 - 1 g(x) = -1 Plug in y = 0 g(x) = -3x - 1 0 = -3x - 1 1 = -3x = x 3 1 3 1 (0, -1) is the y-intercept ( , 0) is the x-intercept *g(x) is the same as y
Find the x and y-intercepts of 6x - 3y =-18 x-intercept y-intercept Plug in y = 0 6x - 3y = -18 6x -3(0) = -18 6x - 0 = -18 6x = -18 x = -3 Plug in x = 0 6x -3y = -18 6(0) -3y = -18 0 - 3y = -18 -3y = -18 y = 6 (-3, 0) is the x-intercept (0, 6) is the y-intercept
Find the x and y-intercepts of x = 3 x-intercept y-intercept Plug in y = 0. A vertical line never crosses the y-axis. There is no y. Why? There is no y-intercept. x = 3 is a vertical line so x always equals 3. (3, 0) is the x-intercept. x y
Find the x and y-intercepts of y = -2 y-intercept x-intercept y = -2 is a horizontal line Plug in y = 0. y cannot = 0 because y = -2. y = -2 is a horizontal line so it never crosses the x-axis. so y always equals -2. (0,-2) is the y-intercept. x There is no x-intercept. y
Graphing Equations Example: Graph the equation -5x + y = 2 Solve for y first. -5x + y = 2 y = 5x + 2 Add 5x to both sides The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane.
Graphing Equations Graph y = 5x + 2 x y
Graphing Equations Graph 4x - 3y = 12 Solve for y first 4x - 3y =12 -3y = -4x + 12 Divide by -3 12 -3 4 3 Subtract 4x from both sides -4 y = x + -3 Simplify y = x 4 4 3 The equation y = x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is . Graph the line on the coordinate plane. 4 3
Graphing Equations 4 3 Graph y = x - 4 x y
Reference teachers.henrico.k12.va.us/math/hcpsal gebra2/Documents/2-2/2006_2_2.ppt
