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Warm-Up1. A 66 ft. board is cut into three pieces. The second piece is1.5 times longer than the first. The third piece is twice aslong as the second. How long is each piece?4. Solve and graph the following: -5< x - 4 <25. Solve and graph the following: -3h < 19 or 7h - 3> 18Class Notes & Practice Problems Section of Notebook 2
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Graphs of y = b Where b is any number y = b are always horizontal lines, and are functions Graphs of x = a Where a is any numberx = a are always vertical lines, andare not functions 4
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Graphs of y = mx y = "a number times x" y = mx lines will always go through the origin, and will be at the angle shown. Graphs of y = mx + c y = "a number times x plus c"y = mx + c are similar in shape to y = mx,but do not go through the origin. 5
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Defining Linear EquationsHOW DO I KNOW IF AN EQUATION IS LINEAR? If an equation is linear, a constantchange in the x-value corresponds to a constant change in the y-value. 6
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Solutions to Linear Equations To ask "Is the ordered pair (1,3) a solution to the equation y = 10x - 7?", is the same as asking, " Is the point (1,3) on the line of y = 10x -7"? How do we know?Determine whether the following ordered pairs are solutions tothe equation y = 10x -7a.) (1,3) b.) (2, 13) c.) (-1,-3) 7
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The slope of the line passing through the twopoints (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 9
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Example: Find the slope of the line passingthrough 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 2 m= = = =1 x2 – x1 4–2 2 y (4, 5) (2, 3) 2 2 x 10
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Finding the x-and y-intercepts of Linear Equations What does it mean to INTERCEPT a pass in football?The path of the defender crosses the path of the thrown football. 11
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The x-intercept is where (2, 0) the graph crosses the x- axis. The y-coordinate is always zero.The y-intercept is where the graph crosses the y- axis. (0, 6) The x-coordinate is always zero. 12
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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. 23
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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. y2. Plot the y-intercept, (0, -4). x change in y 23. The slope is 2. m = = change in x 1 (1, -2)4. Start at the point (0, 4). 2 (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.5. Draw the line through (0, 4) and (1, -2). 24
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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 mpassing through the point (x1, y1).Example: yThe graph of the equation 8 m=- 1 2y – 3 = - 1 (x – 4) is a line 2 4 (4, 3)of slope m = - 1 passing 2through the point (4, 3). x 4 8 25
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Example: Write the slope-intercept form for the equationof 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 26
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Example: Write the slope-intercept form for theequation of the line through the points (4, 3) and (-2,5). 5–3 =- 2 =- 1 Calculate the slope. m= -2 – 4 6 3 y – y1 = m(x – x1) Point-slope form 1 1 y–3=- (x – 4) Use m = - and the point (4, 3). 3 3 y = - 1 x + 13 Slope-intercept form 3 3 27
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Two lines are parallel if they have the same slope.If the lines have slopes m1 and m2, then the linesare parallel whenever m1 = m2. y (0, 4)Example:The lines y = 2x – 3 y = 2x + 4and y = 2x + 4 have slopesm1 = 2 and m2 = 2. x y = 2x – 3The lines are parallel. (0, -3) 28
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Two lines are perpendicular if their slopes arenegative reciprocals of each other.If two lines have slopes m1 and m2, then the linesare perpendicular whenever y 1 m2= - or m1m2 = -1. y = 3x – 1 m1 (0, 4) 1Example: y=- x+4 3The lines y = 3x – 1 and 1y = - x + 4 have slopes 3 xm1 = 3 and m2 = -1 . 3 (0, -1)The lines are perpendicular. 29
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Equations of the form ax + by = c are calledlinear equations in two variables. yThis is the graph of theequation 2x + 3y = 12. x -2 2The point (0,4) is the y-intercept.The point (6,0) is the x-intercept. 30
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