Linear Equations in Two Variables

Linear Equations in Two Variablesmay 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 VariablesConsider the equationThe equations solution set is infinite because there are an infinite number of xs and ys 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-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) + 5y = -8 + 5y = -3x = 4

y = -3

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.

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), }

xy0-4603-23/2-3-3-6-6-8

Graphing a Solution SetTo obtain a more complete picture of a solution set we can graph the ordered pairs from our table onto a rectangular coordinate system.Lets familiarize ourselves with the Cartesian coordinate system.

Cartesian Planex- axisy-axisQuadrant I(+,+)Quadrant II( - ,+)Quadrant IV(+, - )Quadrant III( - , - )origin

Graphing Ordered Pairs on a Cartesian Planex- axisy-axisBegin at the originUse the x-coordinate to move right (+) or left (-) on the x-axisFrom that position move either up(+) or down(-) according to the y-coordinatePlace 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 xy

(3,-2)

(3/2,-3)(-6, -8)

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.

Graphing Linear Equationsin Two VariablesThe 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.

yx

Graphing Linear Equationsin Two VariablesOn our previous graph, y = 2x 3y = 12, find the intercepts.

The x-intercept is (6,0).

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

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

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

Special Lines

y + 5 = 0 x = 3y = -5yy = # is a horizontal line x = # is a vertical line

y

x

x

SLOPE- is the rate of changeWe sometimes think of it as the steepness, slant, or grade.

Slope formula:

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

SlopesPositive slopes rise from left to right

Negative slopes fall from left to right

Special SlopesVertical 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

Slope-Intercept Formy = mx + bwhere m is the slope and b is the y-intercept

Graph using Slope-Intercept form

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

y

1x3

b

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

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

Write an Equationgiven the slope and y-interceptGiven: That a line passes through (0,-9) and has a slope of , write its equation.(0,-9) is the y-intercept (because x=0) is the slope or mPlug into the Slope Intercept Formula to get: y= x - 9

Point-Slope FormAt 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 Point-Slope when you dont have a y-interceptGiven 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.

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.

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