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11-3

Using Slopes and Intercepts. 11-3. Warm Up. Problem of the Day. Lesson Presentation. Pre-Algebra. Problem of the Day Write the equation of a straight line that passes through fewer than two quadrants on a coordinate plane. x = 0 or y = 0. 1. 2. 7. 1. 2. 2. 1. -. 5.

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11-3

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  1. Using Slopes and Intercepts 11-3 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

  2. Problem of the Day Write the equation of a straight line that passes through fewer than two quadrants on a coordinate plane. x = 0 or y = 0

  3. 1 2 7 1 2 2 1 - 5 FOCUS QUESTIONS Find the slope of the line that passes through each pair of points. 1.(3, 6) and (-1, 4) 2. (1, 2) and (6, 1) 3. (4, 6) and (2, -1) 4. (-3, 0) and (-1, 1) SILENT

  4. Put everything you know about solving equations, lines, coordinates, etc. to answer this “problem” 1)Find the x-intercept and y-intercept of the line 4x – 3y = 12. 2) Then use the intercepts to graph the equation in a linear manner. WHAT AM I TALKING ABOUT?

  5. y2–y1 x2–x1 You Should Know By Now! How to simplify variable equations. How us utilize inverse operations How to “clear fractions. How to solve linear equations. How to graph linear equations. How to utilize the Distributive Property. How to get all the variables on one side of the = How to solve inequalities How to graph both equations and inequalities. How to calculate slope of a line. How to use “rise over run” NOW you will memorize, and utilize A new formula: y = mx + b

  6. What is an INTERCEPTION in the sport of football, soccer, basketball? What is meant when one says, ‘I am going to intercept the ball?

  7. “SLOPEINTERCEPT” Learn to use slopes and slope intercepts to graph linear equations.

  8. WRITE THIS DOWN! x-intercept y-intercept slope-intercept form and y = mx + b

  9. You can graph a linear equation easily by finding the x-intercept and the y-intercept. The x-intercept of a line is the value of x where the line crosses the x-axis (where y = 0). The y-intercept of a line is the value of y where the line crosses the y-axis (where x = 0).

  10. 4x = 12 4 4 Put everything you know about solving equations, lines, coordinates, etc. to answer this “problem” Find the x-intercept and y-intercept of the line 4x – 3y = 12. Use the intercepts to graph the equation. Find the x-intercept (y = 0). 4x – 3y = 12 4x – 3(0) = 12 4x = 12 x = 3 The x-intercept is 3.

  11. Additional Example 1 Continued The graph of 4x – 3y = 12 is the line that crosses the x-axis at the point (3, 0) Now, we can solve for y

  12. -3y = 12 -3 -3 Additional Example 1 Continued Find the y-intercept (x = 0). 4x – 3y = 12 4(0) – 3y = 12 –3y = 12 y = –4 The y-intercept is –4.

  13. Additional Example 1 Continued The graph of 4x – 3y = 12 is the line that crosses the x-axis at the point (3, 0) and the y-axis at the point (0, –4).

  14. 8x = 24 8 8 Try This: Example 1 Find the x-intercept and y-intercept of the line 8x – 6y = 24. Use the intercepts to graph the equation. Find the x-intercept (y = 0). 8x – 6y = 24 8x – 6(0) = 24 8x = 24 x = 3 The x-intercept is 3.

  15. -6y = 24 -6 -6 Try This: Example 1 Continued Find the y-intercept (x = 0). 8x – 6y = 24 8(0) – 6y = 24 –6y = 24 y = –4 The y-intercept is –4.

  16. Try This: Example 1 Continued The graph of 8x – 6y = 24 is the line that crosses the x-axis at the point (3, 0) and the y-axis at the point (0, –4).

  17. y = mx + b Slope y-intercept In an equation written in slope-intercept form, y = mx + b, m is the slope and b is the y-intercept.

  18. Helpful Hint For an equation such as y = x – 6, write it as y = x + (–6) to read the y-intercept, –6.

  19. Write each equation in slope-intercept form, and then find the slope and y-intercept. A. 2x + y = 3 2x + y = 3 –2x–2x Subtract 2x from both sides. y = 3 – 2x Rewrite to match slope-intercept form. y = –2x + 3 The equation is in slope-intercept form. m = –2 b= 3 The slope of the line 2x + y = 3 is –2, and the y-intercept is 3.

  20. 5y = x 5 3 3 5 5 y = x + 0 3 m = 5 3 The slope of the line 5y = 3x is , and the y-intercept is 0. 5 Additional Example 2B: Using Slope-Intercept Form to Find Slopes and y-intercepts B. 5y = 3x 5y = 3x Divide both sides by 5 to solve for y. The equation is in slope-intercept form. b = 0

  21. 3y 3 4 –4x 3 3 4 9 3 = + 3 4 y =- x + 3 m =- 3 The slope of the line 4x+ 3y = 9 is – , and the y-intercept is 3. Additional Example 2C: Using Slope-Intercept Form to Find Slopes and y-intercepts C. 4x + 3y = 9 4x + 3y = 9 –4x–4x Subtract 4x from both sides. 3y = 9 – 4x Rewrite to match slope-intercept form. 3y = –4x + 9 Divide both sides by 3. The equation is in slope-intercept form. b= 3

  22. WARM-UP Write each equation in slope-intercept form, and then find the slope and y-intercept. A. 4x + y = 4 –4x–4x Subtract 4x from both sides. y = 4 – 4x Rewrite to match slope-intercept form. y = –4x + 4 The equation is in slope-intercept form. m = –4 b= 4 The slope of the line 4x + y = 4 is –4, and the y-intercept is 4.

  23. 7y = x 7 2 2 7 7 y = x + 0 2 m = 7 2 The slope of the line 7y = 2x is , and the y-intercept is 0. 7 WARM-UP Write this equation in slope-intercept form, and then find the slope and y-intercept. 7y = 2x 7y = 2x Divide both sides by 7 to solve for y. The equation is in slope-intercept form. b = 0

  24. 4y 4 5 –5x 4 4 5 8 4 = + 4 5 y =- x + 2 m =- 4 The slope of the line 5x + 4y = 8 is – , and the y-intercept is 2. Try This: 5x + 4y = 8 5x + 4y = 8 –5x–5x Subtract 5x from both sides. 4y = 8 – 5x Rewrite to match slope-intercept form. 4y =-5x + 8 Divide both sides by 4. The equation is in slope-intercept form. b= 2

  25. y = mx + b • QUESTION: Find the equation of the straight line that has slope m = 4 and passes through the point (–1, –6). They’ve given me the value of the slope; in this case, m = 4. Also, in giving me a point on the line, they have given me an x-value and a y-value for this line: x = –1 and y = –6. In the slope-intercept form of a straight line, I have y, m, x, and b. So the only thing I don't have so far is a value for is b (which gives me the y-intercept). Then all I need to do is plug in what they gave me for the slope and the x and y from this particular point, and then solve for b: y = mx + b(–6) = (4)(–1) + b–6 = –4 + b–2 = b Then the line equation must be "y = 4x – 2".

  26. What if they don't give you the slope? • Find the equation of the line that passes through the points (–2, 4) and (1, 2). • Well, if I have two points on a straight line, I can always find the slope; that's what the slope formula is for. What if they don't give you the slope? Find the equation of the line that passes through the points (–2, 4) and (1, 2). Well, if I have two points on a straight line, I can always find the slope; that's what the slope formula is for.

  27. rise = = – y2 – y1 run = x2 – x1 1 – (–4) 5 5 0 – 3 –5 –3 3 3 = 5 = – 3 Additional Example 2 Continued Choose two points on the line: (0, 1) and (3, –4). Guess by looking at the graph: Use the slope formula. Let (3, –4) be (x1, y1) and (0, 1) be (x2, y2). –5 3

  28. y2 – y1 = x2 – x1 The slope of the given line is – . –4 – 1 5 –5 3 – 0 3 3 = 5 = – 3 Additional Example 2 Continued Notice that if you switch (x1, y1) and (x2, y2), you get the same slope: Let (0, 1) be (x1, y1) and (3, –4) be (x2, y2).

  29. What if they don't give you the slope? • Find the equation of the line that passes through the points (–2, 4) and (1, 2). • Well, if I have two points on a straight line, I can always find the slope; that's what the slope formula is for. What if they don't give you the slope? Find the equation of the line that passes through the points (–2, 4) and (1, 2). Well, if I have two points on a straight line, I can always find the slope; that's what the slope formula is for.

  30. Find the equation of the line that passes through the points (–2, 4) and (1, 2). Now I have the slope and two points. I know I can find the equation (by solving first for "b") if I have a point and the slope. So I need to pick one of the points (it doesn't matter which one), and use it to solve for b. Using the point (–2, 4), I get: y = mx + b4 = (– 2/3)(–2) + b4 = 4/3 + b4 – 4/3 = b12/3 – 4/3 = bb = 8/3 ...so  y = ( – 2/3 ) x + 8/3.

  31. On the other hand, if I use the point (1, 2), I get: y = mx + b2 = (– 2/3)(1) + b2 = – 2/3 + b2 + 2/3 = b6/3 + 2/3 = bb = 8/3 So it doesn't matter which point I choose. Either way, the answer is the same: y = (– 2/3)x + 8/3 As you can see, once you have the slope, it doesn't matter which point you use in order to find the line equation. The answer will work out the same either way.

  32. Additional Example 3: Entertainment Application A video club charges $8 to join, and $1.25 for each DVD that is rented. The linear equation y = 1.25x + 8 represents the amount of money y spent after renting x DVDs. Graph the equation by first identifying the slope and y-intercept. The equation is in slope-intercept form. y = 1.25x + 8 m =1.25 b = 8

  33. Additional Example 3 Continued The slope of the line is 1.25, and the y-intercept is 8. The line crosses the y-axis at the point (0, 8) and moves up 1.25 units for every 1 unit it moves to the right.

  34. Try This: Example 3 A salesperson receives a weekly salary of $500 plus a commission of 5% for each sale. Total weekly pay is given by the equation S = 0.05c + 500. Graph the equation using the slope and y-intercept. The equation is in slope-intercept form. y = 0.05x + 500 m =0.05 b = 500

  35. y 2000 1500 1000 500 x 15,000 10,000 5000 Try This: Example 3 Continued The slope of the line is 0.05, and the y-intercept is 500. The line crosses the y-axis at the point (0, 500) and moves up 0.05 units for every 1 unit it moves to the right.

  36. y2 – y1 = x2 – x1 4 – (–4) 8 –1 – 3 –4 = Additional Example 4: Writing Slope-Intercept Form Write the equation of the line that passes through (3, –4) and (–1, 4) in slope-intercept form. Find the slope. = –2 The slope is –2. Choose either point and substitute it along with the slope into the slope-intercept form. y = mx + b Substitute –1 for x, 4 for y, and –2 for m. 4 = –2(–1) + b 4 = 2 + b Simplify.

  37. Additional Example 4 Continued Solve for b. 4 = 2 + b –2–2 Subtract 2 from both sides. 2 = b Write the equation of the line, using –2 for m and 2 for b. y = –2x + 2

  38. y2 – y1 = x2 – x1 6 – 2 4 2 – 1 1 = Try This: Example 4 Write the equation of the line that passes through (1, 2) and (2, 6) in slope-intercept form. Find the slope. = 4 The slope is 4. Choose either point and substitute it along with the slope into the slope-intercept form. y = mx + b Substitute 1 for x, 2 for y, and 4 for m. 2 = 4(1) + b 2 = 4 + b Simplify.

  39. Try This: Example 4 Continued Solve for b. 2 = 4 + b –4–4 Subtract 4 from both sides. –2 = b Write the equation of the line, using 4 for m and –2 for b. y = 4x – 2

  40. y = – x + 2 3 4 Lesson Quiz Write each equation in slope-intercept form, and then find the slope and y-intercept. 1. 2y – 6x = –10 2. –5y – 15x = 30 Write the equation of the line that passes through each pair of points in slope-intercept form. 3. (0, 2) and (4, –1) 4. (–2, 2) and (4, –4) y = 3x – 5; m = 3; b = –5 y = –3x – 6; m = –3; b = –6 y = –x

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