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Linear Equations in Two Variables

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.

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Linear Equations in Two Variables

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  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 = -8 + 5 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), … }

  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 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)

  11. Graphing Linear Equationsin 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

  12. Graphing Linear Equationsin 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

  13. Finding INTERCEPTS 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 = -4 Thus, the y-intercept is (0,-4). • 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).

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

  15. 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. Slopes 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. 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: • 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 • Are the lines parallel, perpendicular or neither? • Solve for y to get in Slope-Intercept form. • Then compare slopes. Same Slope Parallel Lines

  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 Equationgiven 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. 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

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