Linear Equations 02/11/12 lntaylor ©. Table of Contents Learning Objectives Graph linear equations...

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

Slope Intercept Method

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

Graph y = 2x + 1

Step 1 – Identify information

Identify slope m; Δ’s; b

m = 2 1Δy = 2; Δx = 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

Now you try!

y = 3x - 5

Graph y = 3x - 5

m = 3 1Δy = 3; Δx = 1

b= - 5

Step 2 - Draw

Locate 0,0

Go up or down b

Go up or down Δy

Go right Δx

Connect dots

Label the line

y = 3x - 5

Now you try!

y = - ¾ x + 2

Graph y = - ¾ x + 2

m = - 3 4Δy = - 3; Δx = 4

b = + 2

Step 2 - Draw

Locate 0,0

Go up or down b

Go up or down Δy

Go right Δx

Connect dots

Label the line

y = - ¾x + 2

Now you try a “hard one” !

y = - ¾ x - ½

Graph y = - ¾ x - ½

m = - 3 4Δy = - 3; Δx = 4

b = - ½

Step 2 - Draw

Locate 0,0

Go up or down b

Go up or down Δy

Go right Δx

Connect dots

Label the line

y = - ¾x - ½

Intercepts Method

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

Graph 3x + 4y = 12

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

Locate 0,0

Go right or left xi

Connect dots

Label the line

3x + 4y = 12

Now you try!

2x + 5y = 10

Graph 2x + 5y = 10

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

Locate 0,0

Go right or left xi

Connect dots

Label the line

2x + 5y = 10

Now you try!

-2x - 5y = 10

Graph - 2x - 5y = 10

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

Locate 0,0

Go right or left xi

Connect dots

Label the line

- 2x - 5y = 10

Last one!

2x - 4y = 12

Graph 2x - 4y = 12

Identify yi and xi

yi = 12/- 4or - 3

xi = 12/2 or 6

Step 2 - Draw

Locate 0,0

Go up or down yi

Locate 0,0

Go right or left xi

Connect dots

Label the line

2x - 4y = 12

Reading graphs

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

Now you try!

Put another dot

You have your m

Step 3 - Find b

+ 35 y = x

Now you try a negative slope!

Put another dot

You have your m

Step 3 - Find b

- 83 y = x

- 2 ⅓

Rewriting Equations

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 =

Now you try!

• Given equation - 3x + 5y = 15

Rewrite sign if necessary!

- 3x + 5y = 15- 3x 5 155+ y =

y = 3x + 3 5

Now you try a hard one!

• Given equation - 3x + 5y = - 15

- 3x + 5y = - 15- 3x 5 - 155+ y =

y = 3x - 3 5

Rewriting slope intercept to standard form

• Given equation y = ¾x + 7

Multiply everything by denominator

y = 3x4

+ 74(4)

+ 7y = 3x

- 3x + 4y = 28

Now you try!

• Given equation y = ⅓x - 7

Multiply everything by denominator

y = 1x3

- 73(3)

- 7y = 1x

- x + 3y = - 21

Constructing a line given two points

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

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

___-___

-m = 1 7

y – 2 = 1 (x – – 3) 7

y = 1 x + 2 3 7 7

Now you try!

Find equation (3,2) (- 4,- 5)Step 1 – Identify information

Dot the first point

Start at 0,0; find second point

Put another dot

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

___-___

-m = 1

y – 2 = 1(x - 3)

y = x - 1

Now you try!

(2,4) (3,-2)

y = - 6x + 16

Determining Points on a Line

Is (2,3) on the line y = 2x + 1

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

Is (2,3) on the line y = 2x + 1(2)3 = 2 + 1

Now you try!

y = 2x - 1

Is (2,3) on the line y = 2x - 1

Dot the first 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

Is (2,3) on the line y = 2x - 1(2)3 = 2 - 1

Now you try!

Is the point (-2,5) on the line y = 2x - 1

NO 5 ≠ - 5

• 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

Graph y ≥ 2x - 1

Dot the first point

Put another dot

Draw solid or dotted line

Step 2 – Shade the graph

Turn symbol 1/4 turn left

≥ means solid

Now you try!

Graph y < 2x - 1

Dot the first point

Put another dot

< means dotted

Now you try a hard one!

Graph y < 2

Dot the first point

No slope means no 2nd point

< means dotted

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