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Linear Equations
02/11/12 lntaylor ©
Table of Contents
Learning Objectives
Graph linear equations with slope intercept method
Graph linear equations using intercepts
Reading a graph
Rewrite linear equations from standard to slope intercept form
Constructing a line from 2 points
Determining if a line is on a point
Graphing Inequalities
02/11/12 lntaylor ©
Learning Objectives
TOC02/11/12 lntaylor ©
Linear Equations
• In these sections you will learn/review how to:
– Meet or exceed proficiency in AF 6.0, 7.0 and 8.0
– Pass questions regarding these standards on District or State Tests
– Understand Algebra Functions based on how x values change y values
– Understand Linear Equations vs. Linear Inequalities
TOC02/11/12 lntaylor ©
Slope Intercept Method
TOC02/11/12 lntaylor ©
Linear Equations
• Slope intercept method y = mx + b
– You are given certain information in a slope intercept equation
• Given a slope which is called m
• M is a fraction that denotes Δy / Δx
• Δy informs whether the slope will be + or –
• Δx informs whether the slope is steep or flat
• Given a y intercept (yi or b) which informs where a line crosses the y axis
TOC02/11/12 lntaylor ©
0,0
Graph y = 2x + 1
Step 1 – Identify information
Identify slope m; Δ’s; b
m = 2 1Δy = 2; Δx = 1
b= +1
Step 2 - Draw
Locate 0,0
Go up or down b
Put a dot for the 1st point
Go up or down Δy
Go right Δx
Put a dot for 2nd point
Connect dots
Label the line
y = 2x + 1
TOC02/11/12 lntaylor ©
Now you try!
y = 3x - 5
TOC02/11/12 lntaylor ©
0,0
Graph y = 3x - 5
Step 1 – Identify information
Identify slope m; Δ’s; b
m = 3 1Δy = 3; Δx = 1
b= - 5
Step 2 - Draw
Locate 0,0
Go up or down b
Put a dot for the 1st point
Go up or down Δy
Go right Δx
Put a dot for 2nd point
Connect dots
Label the line
y = 3x - 5
TOC02/11/12 lntaylor ©
Now you try!
y = - ¾ x + 2
TOC02/11/12 lntaylor ©
0,0
Graph y = - ¾ x + 2
Step 1 – Identify information
Identify slope m; Δ’s; b
m = - 3 4Δy = - 3; Δx = 4
b = + 2
Step 2 - Draw
Locate 0,0
Go up or down b
Put a dot for the 1st point
Go up or down Δy
Go right Δx
Put a dot for 2nd point
Connect dots
Label the line
y = - ¾x + 2
TOC02/11/12 lntaylor ©
Now you try a “hard one” !
y = - ¾ x - ½
TOC02/11/12 lntaylor ©
0,0
Graph y = - ¾ x - ½
Step 1 – Identify information
Identify slope m; Δ’s; b
m = - 3 4Δy = - 3; Δx = 4
b = - ½
Step 2 - Draw
Locate 0,0
Go up or down b
Put a dot for the 1st point
Go up or down Δy
Go right Δx
Put a dot for 2nd point
Connect dots
Label the line
y = - ¾x - ½
TOC02/11/12 lntaylor ©
Intercepts Method
TOC02/11/12 lntaylor ©
Linear Equations
• Intercepts method using standard form ax + by = c
– You are given certain information in a standard form equation
• Given a constant c
• X intercept is determined by dividing c/a or the x coefficient
• Y intercept is determined by dividing c/b or the y coefficient
• M is determined by reversing the sign for a and dividing by b or –a/b
TOC02/11/12 lntaylor ©
0,0
Graph 3x + 4y = 12
Step 1 – Identify information
Identify yi and xi
yi = 12/4 or 3
xi = 12/3 or 4
Step 2 - Draw
Locate 0,0
Go up or down yi
Put a dot for the 1st point
Locate 0,0
Go right or left xi
Put a dot for 2nd point
Connect dots
Label the line
3x + 4y = 12
TOC02/11/12 lntaylor ©
Now you try!
2x + 5y = 10
TOC02/11/12 lntaylor ©
0,0
Graph 2x + 5y = 10
Step 1 – Identify information
Identify yi and xi
yi = 10/5 or 2
xi = 10/2 or 5
Step 2 - Draw
Locate 0,0
Go up or down yi
Put a dot for the 1st point
Locate 0,0
Go right or left xi
Put a dot for 2nd point
Connect dots
Label the line
2x + 5y = 10
TOC02/11/12 lntaylor ©
Now you try!
-2x - 5y = 10
TOC02/11/12 lntaylor ©
0,0
Graph - 2x - 5y = 10
Step 1 – Identify information
Identify yi and xi
yi = 10/-5 or - 2
xi = 10/- 2 or - 5
Step 2 - Draw
Locate 0,0
Go up or down yi
Put a dot for the 1st point
Locate 0,0
Go right or left xi
Put a dot for 2nd point
Connect dots
Label the line
- 2x - 5y = 10
TOC02/11/12 lntaylor ©
Last one!
2x - 4y = 12
TOC02/11/12 lntaylor ©
0,0
Graph 2x - 4y = 12
Step 1 – Identify information
Identify yi and xi
yi = 12/- 4or - 3
xi = 12/2 or 6
Step 2 - Draw
Locate 0,0
Go up or down yi
Put a dot for the 1st point
Locate 0,0
Go right or left xi
Put a dot for 2nd point
Connect dots
Label the line
2x - 4y = 12
TOC02/11/12 lntaylor ©
Reading graphs
TOC02/11/12 lntaylor ©
0,0
Find equationStep 1 – Identify information
Starting at far left of the graph find two points where diagonal, horizontal and vertical lines meet
Dot the first intersection
Continue to next intersection
Put another dot
Step 2 – Find the slope
With the 1st dot go up / down and count how many grid lines you pass
Continue to the right until you hot the 2nd dot and count how many grid lines you pass
You have your m
Step 3 - Find b
b is where the diagonal crosses the y axis
+ 54 y = x
+ 3
TOC02/11/12 lntaylor ©
Now you try!
TOC02/11/12 lntaylor ©
0,0
Find equationStep 1 – Identify information
Starting at far left of the graph find two points where diagonal, horizontal and vertical lines meet
Dot the first intersection
Continue to next intersection
Put another dot
Step 2 – Find the slope
With the 1st dot go up / down and count how many grid lines you pass
Continue to the right until you hot the 2nd dot and count how many grid lines you pass
You have your m
Step 3 - Find b
b is where the diagonal crosses the y axis
+ 35 y = x
- 2
TOC02/11/12 lntaylor ©
Now you try a negative slope!
TOC02/11/12 lntaylor ©
0,0
Find equationStep 1 – Identify information
Starting at far left of the graph find two points where diagonal, horizontal and vertical lines meet
Dot the first intersection
Continue to next intersection
Put another dot
Step 2 – Find the slope
With the 1st dot go up / down and count how many grid lines you pass
Continue to the right until you hot the 2nd dot and count how many grid lines you pass
You have your m
Step 3 - Find b
b is where the diagonal crosses the y axis
- 83 y = x
- 2 ⅓
TOC02/11/12 lntaylor ©
Rewriting Equations
TOC02/11/12 lntaylor ©
Rewriting linear equations
• Given equation 2x + 3y = 12
Divide everything by the y coefficient
Move the slope to the other side and do not forget to change the sign!
2x + 3y = 122x 3 123+ y =
-
+ 4
TOC02/11/12 lntaylor ©
Now you try!
TOC02/11/12 lntaylor ©
Rewriting linear equations
• Given equation - 3x + 5y = 15
Divide everything by the y coefficient
Move the slope to the other side and do not forget to change the sign!
Rewrite sign if necessary!
- 3x + 5y = 15- 3x 5 155+ y =
-
+ 3
y = 3x + 3 5
TOC02/11/12 lntaylor ©
Now you try a hard one!
TOC02/11/12 lntaylor ©
Rewriting linear equations
• Given equation - 3x + 5y = - 15
Divide everything by the y coefficient
Move the slope to the other side and do not forget to change the sign!
Rewrite sign if necessary!
- 3x + 5y = - 15- 3x 5 - 155+ y =
-
- 3
y = 3x - 3 5
TOC02/11/12 lntaylor ©
Rewriting slope intercept to standard form
TOC02/11/12 lntaylor ©
Rewriting linear equations
• Given equation y = ¾x + 7
Multiply everything by denominator
Move the slope to the other side and do not forget to change the sign!
Rewrite sign if necessary!
y = 3x4
+ 74(4)
+ 7y = 3x
- +
28
- 3x + 4y = 28
TOC02/11/12 lntaylor ©
Now you try!
TOC02/11/12 lntaylor ©
Rewriting linear equations
• Given equation y = ⅓x - 7
Multiply everything by denominator
Move the slope to the other side and do not forget to change the sign!
Rewrite sign if necessary!
y = 1x3
- 73(3)
- 7y = 1x
- +
- 21
- x + 3y = - 21
TOC02/11/12 lntaylor ©
Constructing a line given two points
TOC02/11/12 lntaylor ©
0,0
Find equation (-3,2) (4,3)Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
Start at 0,0; find second point
Put another dot
Step 2 – Find the slope
Find Δy (y1 – y2)
Find Δx (x1 – x2)
Reduce and you have m
Step 3 - Find equation
(y – y1) = m(x – x1)
Step 4 - Check line
Draw line and check b
2 3
___-___
-3 4
-m = 1 7
y – 2 = 1 (x – – 3) 7
y = 1 x + 2 3 7 7
TOC02/11/12 lntaylor ©
Now you try!
TOC02/11/12 lntaylor ©
0,0
Find equation (3,2) (- 4,- 5)Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
Start at 0,0; find second point
Put another dot
Step 2 – Find the slope
Find Δy (y1 – y2)
Find Δx (x1 – x2)
Reduce and you have m
Step 3 - Find equation
(y – y1) = m(x – x1)
Step 4 - Check line
Draw line and check b
2 - 5
___-___
3 - 4
-m = 1
y – 2 = 1(x - 3)
y = x - 1
TOC02/11/12 lntaylor ©
Now you try!
(2,4) (3,-2)
y = - 6x + 16
TOC02/11/12 lntaylor ©
Determining Points on a Line
TOC02/11/12 lntaylor ©
0,0
Is (2,3) on the line y = 2x + 1
Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
Use slope for 2nd point
Put another dot
Draw line
Step 2 – Find the point
Start at 0,0 ; find given point
Put a dot
Determine if dot is on line
Step 3 – Another way
Substitute point into equation
= yes ≠ no
NO
Is (2,3) on the line y = 2x + 1(2)3 = 2 + 1
3 ≠ 5 the answer is NO TOC02/11/12 lntaylor ©
Now you try!
(2,3)
y = 2x - 1
TOC02/11/12 lntaylor ©
0,0
Is (2,3) on the line y = 2x - 1
Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
Use slope for 2nd point
Put another dot
Draw line
Step 2 – Find the point
Start at 0,0 ; find given point
Put a dot
Determine if dot is on line
Step 3 – Another way
Substitute point into equation
= yes ≠ no
YES
Is (2,3) on the line y = 2x - 1(2)3 = 2 - 1
3 = 3 the answer is Yes TOC02/11/12 lntaylor ©
Now you try!
Is the point (-2,5) on the line y = 2x - 1
NO 5 ≠ - 5
TOC02/11/12 lntaylor ©
Graphing Inequalities
TOC02/11/12 lntaylor ©
Graphing Inequalities
• Need to know the following– Boundary lines are either solid or dotted– ≤ ≥ means solid line; point on line is included in the solution set– <> means dotted line; point on line is not included in solution set– Memorize the following rhyme
• Shoot an arrow• Which line is right• Turn back left• And shade the night
– What it means
• Shoot an arrow – means draw a line• Which line is right – means solid (≤ ≥) or dotted (< >)• Turn back left – means turn ≤≥<> back ¼ turn left• And shade the night – means shade which way the arrow is pointing
TOC02/11/12 lntaylor ©
0,0
Graph y ≥ 2x - 1
Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
Use slope for 2nd point
Put another dot
Draw solid or dotted line
Step 2 – Shade the graph
Turn symbol 1/4 turn left
Shade
≥
≥
≥ means solid
TOC02/11/12 lntaylor ©
Now you try!
TOC02/11/12 lntaylor ©
0,0
Graph y < 2x - 1
Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
Use slope for 2nd point
Put another dot
Draw solid or dotted line
Step 2 – Shade the graph
Turn symbol 1/4 turn left
Shade
<
<
< means dotted
TOC02/11/12 lntaylor ©
Now you try a hard one!
TOC02/11/12 lntaylor ©
0,0
Graph y < 2
Step 1 – Identify information
Start at 0,0; find first point
Dot the first point
No slope means no 2nd point
Draw solid or dotted line
Step 2 – Shade the graph
Turn symbol 1/4 turn left
Shade
<
<
< means dotted
TOC02/11/12 lntaylor ©