graphing linear equations graphing... · 1 intro. to graphing linear equations the coordinate plane...

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Intro. To Graphing Linear Equations

The Coordinate Plane

A. The coordinate plane has 4 quadrants.

B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate

(the ordinate). The point is stated as an ordered pair (x,y).

C. Horizontal Axis is the X – Axis. (y = 0)

D. Vertical Axis is the Y- Axis (x = 0)

Plot the following points:

a) (3,7) b) (-4,5) c) (-6,-1) d) (6,-7)

e) (5,0) f) (0,5) g) (-5,0) f) (0, -5)

y-axis

x-axis

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Slope Intercept Form

Before graphing linear equations, we need to be familiar with slope intercept form. To understand slope

intercept form, we need to understand two major terms: The slope and the y-intercept.

Slope (m):

The slope measures the steepness of a non-vertical line. It is sometimes referred to as the rise over run. It’s

how fast and in what direction y changes compared to x.

y-intercept:

The y-intercept is where a line passes through the y axis. It is always stated as an ordered pair (x,y). The x

coordinate is always zero. The y coordinate can be found by plugging in 0 for the X in the equation or by

finding exactly where the line crosses the y-axis.

What are the coordinates of the y-intercept line pictured in the diagram above? :

Some of you have worked with slope intercept form of a linear equation before. You may remember:

y = mx + b

Using y = mx + b, can you figure out the equation of the line pictured above?:

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Graphing Linear Equations

Graphing The Linear Equation: y = 3x - 5

1) Find the slope: m = 3 m = 3 . = y .

1 x

2) Find the y-intercept: x = 0 , b = -5 (0, -5)

3) Plot the y-intercept

4) Use slope to find the next point: Start at (0,-5)

m = 3 . = ▲y . up 3 on the y-axis

1 ▲x right 1 on the x-axis

(1,-2) Repeat: (2,1) (3,4) (4,7)

5) To plot to the left side of the y-axis, go to y-int. and

do the opposite. (Down 3 on the y, left 1 on the x)

(-1,-8)

6) Connect the dots.

Do Now:

1) y = 2x + 1 2) y = -4x + 5

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3) y = ½ x – 3 4) y= - ⅔x + 2

5) y = -x – 3 6) y= 5x

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Q3 Quiz 1 Review

Find the equation in slope intercept form and graph:

If your printer doesn’t print the graphs, you must use

your own graph paper.

1) y = 4x - 6

2) y = -2x + 7

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3) y = -x - 5

4) y = 5x + 5

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5) y = - ½ x - 7

6) y = ⅗x - 4

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7) y = ⅔x

8) y = - ⅓x + 4

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Finding the equation of a line in slope intercept form

(y=mx + b)

Example: Using slope intercept form [y = mx + b]

Find the equation in slope intercept form of the line formed by (3,8) and (-2, -7).

A. Find the slope (m): B. Use m and one point to find b:

m = y2 – y1 y = mx + b

x2 – x1 m= 3 x= 1 y= 2

m = (-7) – (8) . 2 = 3(1) + b

(-2) – (3) 2 = 3 + b

-3 -3

m = -15 . -1 = b

-5

m= 3 y = 3x – 1

Example: Using point slope form [ y – y1 = m(x – x1) ]

Find the equation in slope intercept form of the line formed by (3,8) and (-2, -7).

A. Find the slope (m): B. Use m and one point to find b:

m = y2 – y1 y – y1 = m(x – x1)

x2 – x1 m= 3 x= 1 y= 2

y – (2) = 3(x – (1))

m = (-7) – (8) . y – 2 = 3x - 3

(-2) – (3) +2 +2

m = -15 . y = 3x – 1

-5

m= 3

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Find the equation in slope intercept form of the line formed by the given points. When you’re finished, graph

the equation on the give graph.

1) (4,-6) and (-8, 3)

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2) (4,-3) and (9,-3) 3) (7,-2) and (7, 4)

III. Special Slopes

A. Zero Slope B. No Slope (undefined slope)

* No change in Y * No change in X

* Equation will be Y = * Equation will be X =

* Horizontal Line * Vertical Line

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Point-Slope Form y – y1 = m(x – x1)

Slope Intercept Form y = mx + b “y” is by itself

Standard Form: Ax + By = C Constant (number) is by itself

Given the slope and 1 point, write the equation of the line in: (a) point-slope form,

(b) slope intercept form, and (c) standard form:

Example: m = ½ ; (-6,-1)

a) Point-Slope Form b) Slope intercept form c) Standard Form

1) m = -2; (-3,1)

a) Point-Slope Form b) Slope intercept form c) Standard Form

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2) m = - ¾ ; (-8, 5)

Point-Slope Form b) Slope intercept form c) Standard Form

3) m = ⅔; (-6, -4)


4) m = -1 (5, -1)


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Find equation in slope intercept form and graph:

1) (3,-2)(-6,-8) 2) (-6,10) (9,-10)

3) (3,7) (3,-7) 4) (7,-6)(-3,4)

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5) (5,-9)(-5,-9) 6) m= 4 (-2,-5)

7) m= ⅔ (-6,-7) 8) m= -

(8,-1)

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9) m = 0 (4,3) 10) m = undefined (-6, 5)

11) 16x -4y =36 12) 8x+24y = 96

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13) y+7=2(x+1) 14) y+5=(2/5)(x+10)

15) y-7= ¾ (x-12) 16) y-2=-3(x-2)

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Q3 Quiz 2 Review




1) y - 2 = -3(x – 1)

2) 14x + 21y = -84

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3) y + 10 = 5(x + 2)

4) y – 7 = ¼ (x – 20)

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5) 8x – 8y = 56

6) y + 6 = -1(x – 4)

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7) 18x – 12y = -12

8) y – 15 = (-5/3)(x + 9)

Answers: 1) y = -3x + 5 2) y = - ⅔ x - 4 3) y = 5x 4) y = ¼ x + 2

5) y = x - 7 6) y = - x – 3 7) y = (3/2)x - 1 8) y = -(5/3)x

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Graph both of the lines on the same set of axis:

y = -2x + 6 y = -2x – 5

IV. Parallel and Perpendicular Lines:

A. Parallel Lines

* Do not intersect

* Have same slopes

For the given line, find a line that is parallel and passes through the given point and graph

Given Line: Parallel: Given Line: Parallel:

7) y = ⅓ x + 4 (6,1) 8) y = 4x – 5 (2,13)

Given Line: Parallel: Given Line: Parallel:

9) y = -⅔ x + 2 (-9,2) 10) y = –5x + 6 (4,-27)

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Practice Problems: a) Use the two points to find the equation of the line.

b) For the line found in part a, find a line that is parallel and passes through the given

point.

c) Graph both lines on the same set of axis.

Given Line: Parallel:

1) (-5, 13) (3, -3) (4,-10)


2) (-6,0) (3,6) (6,3)

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3) (2,6)(-3,-19) (5,30)


4) (-4,3) (-8,6) (-4, 10)

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5) (2,-5) (-2, -5) (8,-2)


6) (-9,-11)(6,9) (-3,-9)

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7) (8,-3) (-4,9) (-2, 1)


8) (3,6)(3,-6) (7,-3)

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9) (4,-3)(-6,-8) (6,7)


10) (2,4)(-6,-12) (-3,-5)

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11) Find the equation of the line parallel to y = 3x – 2, passing through (-2, 1).

12) Find the equation of the line parallel to y = -½x – 5, passing through (-2, 7)

13) Find the equation of the line parallel to y = -¼ x + 2, passing through (-8, 4)

14) Find the equation of the line parallel to y = (3/2)x + 6, passing through (-6, -11)

15) Find the equation of the line parallel to y = -5, passing through (2,7)

16) Find the equation of the line parallel to x = 5, passing through (6, -4).

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Q3 Quiz 3 Review

FOLLOW REQUIRED FORMAT AND SHOW ALL PROPER WORK!

a) Use the two points to find the equation of the line.

b) For the line found in part a, find a line that is parallel and passes through the given point.

c) Graph both lines on the same set of axis.


1) (-4, 13) (3, -8) (4,-17)


2) (8,1) (-4,-5) (-6,2)

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3) (5,4) (-4,4) (-6,-7)

For #’s 4-7, just find the equation. You do not have to graph.

4) Find the equation of the line parallel to y = -⅗x – 2, passing through (-5, 7).

5) Find the equation of the line parallel to y = 4x – 5, passing through (-4, 9)

6) Find the equation of the line parallel to y = 2, passing through (-8, -9)

7) Find the equation of the line parallel to x = 5, passing through (-6, -11)

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Solving Systems of Equations

Graphically

A system of equations is a collection of two or more equations with a same set of unknowns. In solving a

system of equations, we try to find values for each of the unknowns that will satisfy every equation in the

system. When solving a system containing two linear equations there will be one ordered pair (x,y) that will

work in both equations.

To solve such a system graphically, we will graph both lines on the same set of axis and look for the point of

intersection. The point of intersection will be the one ordered pair that works in both equations. We must then

CHECK the solution by substituting the x and y coordinates in BOTH ORIGINAL EQUATIONS.

1) Solve the following system graphically:

y = 2x – 5

y = - ⅓x + 2

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Solve each of the systems of equations graphically:

2) y + 1 = -3(x – 1) 7x + 7y = 42

3) y – 9 = ¾ (x – 12) 6x + 12y = -60

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4) 12x – 8y = 48

y – 4 = -2(x – 2)

5) y + 5 = 2(x + 4) y – 10 = - ½ (x + 4)

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Solve each system graphically and check:

6) y = -4x -5

y = 2x -7

7) 6x + 3y =21

12x + 16y = -48

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8) 12x – 6y = -6

16x -8y = 40

9) y= -4

x = 7

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10) y-2= (3/5)(x-10)

y+11 =2(x+7)

11) 6x + 9y = 45

9x +15y = 75

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12) x = 5

y-12 = -3(x+2)

13) 9x – 18y = 126

y = -4

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Q3 Quiz 4 Review 1) y = 2x + 6

y = - ½x - 4

2) 15x - 15y = -45

y - 2 = -3(x + 1)

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3) y + 3 = (-2/3)(x - 3)

y + 1 = -1(x + 2)

4) 24x - 18y = -18

y = -7

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5) y + 2 = (-3/4)(x - 8)

17x - 34y = 204

6) x = -6

y + 15 = (5/3)(x + 9)

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7) y - 5 = ¼(x - 4)

45x - 15y = 105

8) 24x - 12y = -72

y – 2 = 2(x + 2)

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9) 11x + 44y = -176

y - 3 = - ¼ (x + 4)

10) y + 4 = (2/3)(x + 12)

25x + 50y = -150

Answer Key:

1) (-4,-2) 2) (-1,2) 3) (-6,3) 4) (-6, -7) 5) (8,-2)

6) (-6,-10) 7) (4,5) 8) Many Solutions 9) No Solution 10) (-6,0)

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Graphing Inequalities

When we solved and graphed inequalities with only one variable (ex: x > 3), we moved on to compound

inequalities (AND/OR). We would graph both inequalities on the same number line and decide what to keep

based on whether it was an AND or an OR problem. When we graphed linear equations on the coordinate plane

we moved on to solving systems of equations graphically.

When we graph inequalities in two variables on the coordinate plane, we do not graph compound inequalities.

We move on to solving systems of inequalities. It takes a little from both inequalities with one variable and

solving systems graphically.

Graph the Inequality:

y > ¼ x + 3

Step 1: Graph the line.

y > ¼ x + 3

m = ¼ = ▲y = up 1

▲x r 4

y-int= (0,3)

Step 2: Test a point one up from the from

the y-int and one down from the y-int):

(0, 2) (0, 4)

2 > ¼ (0)+3 4 > ¼ (0) + 3

2 > 3 4 > 3

FALSE TRUE

Step 3: Shade towards the “true” point (0,4)

When you “test”, you must do

it in the original inequality!

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1) 6x - 9y > 36

2) y - 3 > -2(x + 1)

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3) 12x + 9y < 27

4) y + 4 > -3(x - 3)

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5) y > 4

6) x < -6

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Q3 Quiz 5 Review

1) 72x – 216y < -432

2) y + 1 > ⅖ (x + 10)

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3) y – 5 < - ½ (x + 10)

4) 48x + 12y < -48

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5) x > 7

6) y < -2

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7) x < -4

8) y > 6

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Graphing Systems of Inequalities Solve the system of inequalities graphically: y > ¼ x + 3

y < 3x – 5

Step 1: Graph the 1st inequality Step 2: Graph the 2

nd inequality

y > ¼ x + 3 y < 3x – 5

m = ¼ = m = 3/1 =

y-int= (0,3) y-int.= (0,-5)

(0, 2) TEST (0, 4) (0-6) TEST (0,-4)

2 > ¼ (0)+3 -3 > ¼ (0) + 3 -6 < 3(0) - 5 -4 < 3(0) - 5

2 > 3 4 > 3 -6 < -5 -4 < -5

FALSE TRUE TRUE FALSE

Step 3: Label the area where the shading intersects with an “S”

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2) y - 3 < - ⅓(x – 6)

12x – 6y > -12

3) x > 4

y < -5

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4) 24x + 6x > -6

y > 2

5) y – 6 < ⅔(x - 9)

x < -3

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Q3 Quiz 6 Review

1) y > 5

3x - y > -3

2) y + 6 > -½ (x - 8)

y - 4 > 2(x – 2)

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3) 15x – 45y < 90

x > 3

4) 21x – 7y > 14

y- 3 > -¼ (x + 12)

graphing linear equations graphing... · 1 intro. to graphing linear equations the coordinate plane...

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