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

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 + 5 y = -3 x = 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), … } x y -4 6 3 -2 3/2 -3 -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 y-axis Quadrant II Quadrant I ( - ,+) (+,+) x- axis origin Quadrant IV (+, - ) Quadrant III ( - , - )

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)

Graphing More Ordered Pairs from our Table for the equation y 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)

Graphing Linear Equations in Two Variables y 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

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

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).

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

Slope formula: SLOPE SLOPE- is the rate of change We 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).

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

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

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

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.

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.

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

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!

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.

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