1. Linear Equations in Two Variables

2. 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.

3. 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?
Ordered Pairs
are listed with the
x-value first and the y-value
second.

4. 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 =
y = -3
x = 4
y = -3

5. Functions
Function- a relationship between two variables (equation) so that for every INPUT there is EXACTLY one OUTPUT.
To determine (algebraically) if an equation is a function we can examine its x/y table. If it is possible to get two different outputs for a certain input- it is NOT a function. In this case an x-value in the table or ordered pairs would repeat.
This may be determined (graphically) by using the Vertical Line Test. If any vertical line would touch the graph at more than one point- it is NOT a function.

6. Using Tables to List Solutions
For an equation
we can list some solutions in a table.
Or, we may list the
solutions in ordered pairs .
{(0,-4), (6,0), (3,-2),
( 3/2, -3), (-3,-6),
(-6,-8), … } x y -4 6 3 -2
3/2 -3 -6 -8 …

7. 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.

8. Cartesian Plane y-axis Quadrant II Quadrant I ( - ,+) (+,+) x- axis
origin
Quadrant IV
(+, - )
Quadrant III
( - , - )

9. Graphing Ordered Pairs on a Cartesian Plane
y-axis
Begin at the origin
Use the x-coordinate to move right (+) or left (-) on the x-axis
From that position move either up(+) or down(-) according to the y-coordinate
Place a dot to indicate a point on the plane
Examples: (0,-4)
(6, 0)
(-3,-6)
(6,0)
x- axis
(0,-4)
(-3, -6)

10. Graphing More Ordered Pairs from our Table for the equationy
Plotting more points
we see a pattern.
Connecting the points
a line is formed.
We indicate that the
pattern continues by placing
arrows on the line.
Every point on this line is a
solution of its equation. x
(3,-2)
(3/2,-3)
(-6, -8)

11. Graphing Linear Equations in Two Variablesy
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.
On our previous graph, y = 2x – 3y = 12, find the intercepts. x

12. Graphing Linear Equations in Two Variablesy
On our previous graph, y = 2x – 3y = 12, find the intercepts.
The x-intercept is (6,0).
The y-intercept is (0,-4). x

13. Finding INTERCEPTS
To find the x-intercept: Plug in ZERO for y and solve for x.
2x – 3y = 12
2x – 3(0) = 12
2x = 12
x = 6
Thus, the x-intercept is (6,0).
To find the y-intercept: Plug in ZERO for x and solve for y.
2(0) – 3y = 12
-3y = 12
y = -4
Thus, the y-intercept is (0,-4).

14. Special Lines y + 5 = 0 x = 3 y = -5
y = # is a horizontal line x = # is a vertical line

15. Slope formula: SLOPE SLOPE- is the rate of change
We sometimes think of it as the steepness, slant, or grade.
Slope formula:

16. Slope: Given 2 colinear points, find the slope.
Find the slope of the line containing (3,2) and (-1,5).

17. Positive slopes rise from left to right
Negative slopes fall from left to right

18. 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

19. where m is the slope and b is the y-intercept
Slope-Intercept Form
y = mx + b
where m is the slope and b is the y-intercept

20. Graph using Slope-Intercept form
Given: 2y= 6x – 4
y = 3x – 2
Plot (0, -2) then use 3/1 as rise/run to get 2nd point: y 1 x 3 b
Solve for y.
Plot b on the y-axis.
Use
to plot a second point.
4. Connect the points to make a line.
Rise: positive means UP/ negative means DOWN
Run: positive means RIGHT/ negative means LEFT

21. Determine the relationship between lines using their slopes
Same Slope
Parallel Lines
Are the lines parallel, perpendicular or neither?
Solve for y to get in Slope-Intercept form.
Then compare slopes.

22. Determine the relationship between lines using their slopes
Perpendicular Lines
Are the lines parallel, perpendicular or neither?
Solve for y to get in Slope-Intercept form.
Then compare slopes.

23. Write an Equation given the slope and y-intercept
Given: That a line passes through (0,-9) and has a slope of ½ , write its equation.
(0,-9) is the y-intercept (because x=0)
½ is the slope or m
Plug into the Slope Intercept Formula to get: y= ½ x - 9

24. with a slope of m is given by
Point-Slope Form
At times we may not know the y-intercept. Thus, we need a new formula. The point-slope form of a line going through
with a slope of m is given by
Use the Parentheses!

25. Use Point-Slope when you don’t have a y-intercept
Given two points (1,5) and (-4,-2), write the equation for their line.
Choose one point to plug in for (x1,y1)
Find the slope using both points and the slope formula.
Solve the equation for y.

26. Modeling Data with Linear Equations
Data can sometimes be modeled by a linear function.
Notice there is a basic trend. If we place a line over the tops of the bars it “roughly fits.” Each bar is close to the line. Thus points on the line should estimate our data.
Given the equation to the line we can make predictions about this data.

27. Modeling Data with Linear Equations
The number of U.S. children (in thousands) educated at home for selected years is given in the table. Letting x=3 represent the year 1993, use the first and last data points to write an equation in slope-intercept form to fit the data.
y=128x + 204