Linear functions any_two_points

Linear Equations in Two Variables

Linear Equations in Two Variables

may be put in the form

Ax + By = C,

Where A, B, and C are real numbers and A and B are not

both zero.

Solutions to Linear Equations in Two Variables

Consider the equation The equation’s solution set is

infinite because there are an infinite number of x’s and y’s that make it TRUE.

For example, the ordered pair (0, 10) is a solution because

Can you list other ordered pairs that satisfy this equation?

5 2 20x y

5 0 2 10 20 5 2 20x y

Ordered Pairs are listed with the

x-value first and the y-valuesecond.

Input-Output Machines

We can think of equations as input-output machines. The x-values being the “inputs” and the y-values being the “outputs.”

Choosing any value for input and plugging it into the equation, we solve for the output.

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

y = -8 + 5y = -3

x = 4 y = -3

Using Tables to List Solutions

For an equation we can list some solutions in a table.

Or, we may list thesolutions in ordered

pairs .{(0,-4), (6,0), (3,-2), ( 3/2, -3), (-3,-6), (-6,-8), … }

2 3 12x y x y

0 -4

6 0

3 -2

3/2 -3

-3 -6

-6 -8

… …

Graphing a Solution Set

To obtain a more complete picture of a solution set we can graph the ordered pairs from our table onto a rectangular coordinate system.

Let’s familiarize ourselves with the Cartesian coordinate system.

Cartesian Plane

x- axis

y-axis

Quadrant I(+,+)

Quadrant II( - ,+)

Quadrant IV(+, - )

Quadrant III( - , - )

origin

Graphing Ordered Pairs on a Cartesian Plane

x- axis

y-axis

1) Begin at the origin

2) Use the x-coordinate to move right (+) or left (-) on the x-axis

3) From that position move either up(+) or down(-) according to the y-coordinate

4) Place a dot to indicate a point on the plane

Examples: (0,-4)

(6, 0)

(-3,-6)

(6,0)

(0,-4)(-3, -6)

Graphing More Ordered Pairs from our

Table for the equation

x

y

(3,-2)

(3/2,-3)

(-6, -8)

2 3 12x y •Plotting more points

we see a pattern.

•Connecting the pointsa line is formed.

•We indicate that thepattern continues by placing

arrows on the line.

•Every point on this line is asolution of its equation.

Graphing Linear Equationsin Two Variables

The graph of any linear equation in two variables is a straight line.

Finding intercepts can be helpful when graphing.

The x-intercept is the point where the line crosses the x-axis.

The y-intercept is the point where the line crosses the y-axis.

y

x

Graphing Linear Equationsin Two Variables

On our previous graph, y = 2x – 3y = 12, find the intercepts.

The x-intercept is (6,0).

The y-intercept is (0,-4).

y

x

Finding INTERCEPTS

To find the To find the x-intercept: Plug in x-intercept: Plug in ZERO for y and solve ZERO for y and solve for x.for x.

2x – 3y = 122x – 3y = 12

2x – 3(0) = 122x – 3(0) = 12

2x = 122x = 12

x = 6x = 6Thus, the x-intercept is (6,0).

To find the To find the y-intercept: Plug in y-intercept: Plug in ZERO for x and ZERO for x and solve for y.solve for y. 2(0) – 3y = 122(0) – 3y = 12 2(0) – 3y = 122(0) – 3y = 12

-3y = 12-3y = 12 y = -4y = -4

Thus, the y-intercept is Thus, the y-intercept is (0,-4). (0,-4).

Special Lines y + 5 = 0 x = 3y = -5

y

x

x

y

y = # is a horizontal line x = # is a vertical line

SLOPE- is the rate of change

We sometimes think of it as the steepness, slant, or grade.

2 1

2 1

y y y riseslope m

x x x run

Slope formula:

Slope:Given 2 colinear points, find the slope.

Find the slope of the line containing (3,2) and (-1,5).

2 1

2 1

2 5 3

3 1 4

y ym

x x

Slopes

Positive slopes rise from left to right

Negative slopes fall from left to right

Special Slopes

Vertical lines have UNDEFINED slope (run=0 --- undefined)

Horizontal lines have zero slope (rise = 0)

Parallel lines have the same slope (same slant)

Perpendicular lines have opposite reciprocal slopes

0m

12

1m

m

1 2m m

m undefined

Read 2x:

Bring out 1 whole sheet of paper:

copy and answer

With solution..

SeatworkFind the slope of the

following:

1. (3,2),(5,6)

2. (1,-2),(-2,0)

3. (4,6),(3,0)

4. (-9,6),(-10,3)

5. (-5,9),(-3,6)

6. (-4,8),(6,-1)

7. (0,2),(3,-2)

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