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# January 9, 2014

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### January 9, 2014

1. 1. 1. A 72 ft. pipe is cut into two pieces of lengths in a 5:7 ratio. What are the lengths of the two pieces? 2. A \$40.00 shirt has been discounted 35%. What is the sale price of the shirt? 3. Find a number so that 20 more than one-third of the number equals three-fourths of that number. 4. x + 2 + x - 1 = 2 3 2 5. x + 3 = x - 2 6 4
2. 2. Formulas: The Slope of a Line: The Slope Intercept Form: The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula A linear equation written in the form y = mx + b is in slope-intercept form.
3. 3. Equations and their Graphs: Equations of the form ax + by = c are called linear equations in two variables. y Graph the equation 2x + 3y = 12. Find the x - intercept. Find the y-intercept. x -2 2 4
4. 4. 5
5. 5. The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula y (x2, y2) y2 – y1 change in y (x1, y1) x2 – x1 change in x x 6
6. 6. Example: Find the slope of the line passing through the points (2, 3) and (4, 5). Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5. y2 – y1 5–3 m= = x2 – x1 4–2 2 =1 = 2 y (4, 5) 2 (2, 3) 2 x 7
7. 7. The Slope Intercept Form of a Line: A linear equation written in the form y = mx + b is in slope-intercept form. The slope is m and the y-intercept is (0, b). To graph an equation in slope-intercept form: 1. Write the equation in the form y = mx + b. Identify m and b. 2. Plot the y-intercept (0, b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope. 8
8. 8. Example: Graph the line y = 2x – 4. 1. The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4. y 2. Plot the y-intercept, (0, - 4). x 3. The slope is 2. m = change in y 2 = 1 change in x 4. Start at the point (0, 4). (0, - 4) Count 1 unit to the right and 2 units up 1 to locate a second point on the line. The point (1, -2) is also on the line. (1, -2) 2 5. Draw the line through (0, 4) and (1, -2). 9
9. 9. The Slope Intercept Form of a Line: Example 2: Graph the line 3y - 3x = 6. Last Example: Graph the line -2x + y = -4. slope intercept y = 2x - 4. When you know how to 'decode' the slope-intercept form, you won't have to make a table of x and y values any longer.
10. 10. Class Work: See Handout
11. 11. A linear equation written in the form y – y1 = m(x – x1) is in point-slope form. The graph of this equation is a line with slope m passing through the point (x1, y1). Example: The graph of the equation y 8 m=- y – 3 = - 1 (x – 4) is a line 2 of slope m = - 1 passing 2 through the point (4, 3). 4 1 2 (4, 3) x 4 8 14
12. 12. Example: Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3. Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1) = (-2, 5). y – y1 = m(x – x1) Point-slope form y – y1 = 3(x – x1) Let m = 3. y – 5 = 3(x – (-2)) Let (x1, y1) = (-2, 5). y – 5 = 3(x + 2) Simplify. y = 3x + 11 Slope-intercept form 15
13. 13. Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (2, 5). 5–3 =- 2 =- 1 m= -2 – 4 6 3 y – y1 = m(x – x1) 1 (x – 4) 3 y = - 1 x + 13 3 3 y–3=- Calculate the slope. Point-slope form Use m = - 1 and the point (4, 3). 3 Slope-intercept form 16
14. 14. Two lines are parallel if they have the same slope. If the lines have slopes m1 and m2, then the lines are parallel whenever m1 = m2. y (0, 4) Example: The lines y = 2x – 3 y = 2x + 4 and y = 2x + 4 have slopes m1 = 2 and m2 = 2. x y = 2x – 3 The lines are parallel. (0, -3) 17
15. 15. Two lines are perpendicular if their slopes are negative reciprocals of each other. If two lines have slopes m1 and m2, then the lines are perpendicular whenever y 1 m2= or m1m2 = -1. y = 3x – 1 m1 (0, 4) Example: The lines y = 3x – 1 and 1 y = - x + 4 have slopes 3 m1 = 3 and m2 = -1 . 3 1 y=- x+4 3 x (0, -1) The lines are perpendicular. 18
16. 16. • Linear Equations form straight lines. How do we determine if an equation is linear? It can be rewritten in the form: Ax + By = C This is the Standard Form of a linear equation where: a.) A and B are not both zero. b.) The largest exponent is not greater than 1 Determine Whether the Equations are Linear: 1. 4 - 2y = 6x 2. -4/5x = -2 3. -6y + x = 5y - 2 Remember: This is to determine whether an equation is linear (forms a straight line) or not. The standard form is also used to determine x and y intercepts.
17. 17. Class Work: 8-3: 28-36; Plot & Label 3 points per problem 4.3-- All 20