LINEAR EQUATIONS PART I

1. Basic Coordinate Plane Info2. Review on Plotting Points

3. Finding Slopes4. x and y intercepts

5. Slope-Intercept Form of a Line6. Graphing Lines

7.Determine the equation of a line given two points, slope and one point, or a graph.

Assignments

COORDINATE PLANE

Parts of a plane1.X-axis2.Y-axis3.Origin4.Quadrants I-IV

X-axis

Y-axis

Origin ( 0 , 0 )

QUAD IQUAD II

QUAD III QUAD IV

PLOTTING POINTSRemember when plotting points you always start at the origin. Next you go left (if x-coordinate is negative) or right (if x-coordinate is positive. Then you go up (if y-coordinate is positive) or down (if y-coordinate is negative)

Plot these 4 pointsA (3, -4), B (5, 6), C (-4, 5) and D (-7, -5)

A

BC

D

Graphing Linear Equations

In Slope-Intercept Form

We have already seen that linear equations have two variables and when we plot all the (x,y) pairs that make the equation true we get a line.

In this section, instead of making a table, evaluating y for each x, plotting the points and making a line, we will use The Slope-Intercept Form of the equation to graph the line.

y 2x 1

y x 4

y 32

x 2

These equations are all in Slope-Intercept Form:

Notice that these equations are all solved for y.

Just by looking at an equation in this form, we can draw the line (no tables).

•The constant is the y-intercept.

•The coefficient is the slope.

y 2x 1

y x 4

y 3

2x 2

Constant = 1, y-intercept = 1.

Coefficient = 2, slope = 2.

Constant = -4, y-intercept = -4.

Coefficient = -1, slope = -1.

Constant = -2, y-intercept = -2.

Coefficient = 3/2, slope = 3/2.

The formula for Slope-Intercept Form is:

y mx b;• ‘b’ is the y-intercept.

• ‘m’ is the slope.

On the next three slides we will graph the three equations:

y 2x 1, y x 4, y 3

2x 2

using their y-intercepts and slopes.

y 2x 1

1) Plot the y-intercept as a point on the y-axis. The constant, b = 1, so the y-intercept = 1.

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 2, so the slope = 2/1.

up 2

right 1 up 2

right 1

1) Plot the y-intercept as a point on the y-axis. The constant, b = -4, so the y-intercept = -4.

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -1, so the slope = -1/1.

right 1down 1

right 1

y x 4

down 1

1) Plot the y-intercept as a point on the y-axis. The constant, b = -2, so the y-intercept = -2.

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 3/2, so the slope = 3/2.

right 2

up 3

y 3

2x 2

right 2

up 3

Sometimes we must solve the equation for y before we can graph it.

2x y 3

2x y ( 2x) ( 2x) 3

y 2x 3

The constant, b = 3 is the y-intercept.

The coefficient, m = -2 is the slope.

1) Plot the y-intercept as a point on the y-axis. The constant, b = 3, so the y-intercept = 3.

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -2, so the slope = -2/1.

right 1down 2

right 1

down 2

y 2x 3

SLOPESlope is the ratio of the vertical rise to the horizontal run between any two points on a line. Usually referred to as the rise over run. Slope triangle between two

points. Notice that the slope triangle can be drawn two different ways.

Rise is -10 because we went down

Run is -6 because we went to the left

3

5

6

10

iscasethisinslopeThe

Rise is 10 because we went up

Run is 6 because we went to the right

3

5

6

10iscasethisinslopeThe

Another way to find slope

FORMULA FOR FINDING SLOPE

21

21

12

12

YY

XX

YY

XX

RUN

RISESLOPE

The formula is used when you know two points of a line.

),(),( 2211 YXBandYXAlikelookThey

EXAMPLE

Find the slope of the line between the two points (-4, 8) and (10, -4)

If it helps label the points. 1X 1Y2X 2Y

Then use the formula

12

12

YY

XX

)8()4(

)4()10(

FORMULAINTOSUBSTITUTE

6

7

12

14

)8(4

410

)8()4(

)4()10(

SimplifyThen

X AND Y INTERCEPTSThe x-intercept is the x-coordinate of a point where the graph crosses the x-axis.

The y-intercept is the y-coordinate of a point where the graph crosses the y-axis.

The x-intercept would be 4 and is located at the point (4, 0).

The y-intercept is 3 and is located at the point (0, 3).

SLOPE-INTERCEPT FORM OF A LINEThe slope intercept form of a line is y = mx + b, where “m” represents the slope of the line and “b” represents the y-intercept.

When an equation is in slope-intercept form the “y” is always on one side by itself. It can not be more than one y either.

If a line is not in slope-intercept form, then we must solve for “y” to get it there.

Examples

IN SLOPE-INTERCEPT NOT IN SLOPE-INTERCEPT

y = 3x – 5 y – x = 10

y = -2x + 10 2y – 8 = 6x

y = -.5x – 2 y + 4 = 2x

Put y – x = 10 into slope-intercept form

Add x to both sides and would get y = x + 10

Put 2y – 8 = 6x into slope-intercept form.

Add 8 to both sides then divide by 2 and would get y = 3x + 4

Put y + 4 = 2x into slope-intercept form.

Subtract 4 from both sides and would get y = 2x – 4.

GRAPHING LINESBY MAKING A TABLE OR USING THE

SLOPE-INTERCEPT FORM

I could refer to the table method by input-output table or x-y table. For now I want you to include three values in your table. A negative number, zero, and a positive number.

Graph y = 3x + 2INPUT (X) OUTPUT (Y)

-2 -4

0 2

1 5

By making a table it gives me three points, in this case (-2, -4) (0, 2) and (1, 5) to plot and draw the line.

See the graph.

Plot (-2, -4), (0, 2) and (1, 5)

Then draw the line. Make sure your line covers the graph and has arrows on both ends. Be sure to use a ruler.

Slope-intercept graphing

Slope-intercept graphingSteps1.Make sure the equation is in slope-intercept form.2.Identify the slope and y-intercept.3.Plot the y-intercept.4.From the y-intercept use the slope to get another point to draw the line.

1. y = 3x + 22. Slope = 3 (note that this means the

fraction or rise over run could be (3/1) or (-3/-1). The y-intercept is 2.

3. Plot (0, 2)4. From the y-intercept, we are going

rise 3 and run 1 since the slope was 3/1.

FIND EQUATION OF A LINE GIVEN 2 POINTS

1. Find the slope between the two points.

2. Plug in the slope in the slope-intercept form.

3. Pick one of the given points and plug in numbers for x and y.

4. Solve and find b.5. Rewrite final form.

Find the equation of the line between (2, 5) and (-2, -3).

1. Slope is 2.2. y = 2x + b3. Picked (2, 5) so

(5) = 2(2) + b4. b = 15. y = 2x + 1

Two other ways

Steps if given the slope and a point on the line.1.Substitute the slope into the slope-intercept form.2.Use the point to plug in for x and y.3.Find b.4.Rewrite equation.

If given a graph there are three ways.

One way is to find two points on the line and use the first method we talked about.

Another would be to find the slope and pick a point and use the second method.

The third method would be to find the slope and y-intercept and plug it directly into y = mx + b.

Assignments

Pages 206-208 #’s 1-31, 33-39Pages 233-235 #’s 10-21, 23-36, 43, 44Pages 272-275 #'s 4-39, 54-73 Pages 282-283 #’s 20-34, 49-51Pages 213-215 #'s 3-48 Pages 219-221 #’s 1-32, 35-42Pages 225-227 #'s 18-32, 39-47 Pages 246-249 #'s 3-48, 67-78 Pages 288-290 #'s 3-35, 41-48