0 Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not both zero. Then an equation that can be written in the form: ax + by = c is called a linear equation in two variables If a, b, and c happen to integers with a > 0, then we call this the standard form of a linear equation. A linear equation in two variables is said to be written in standard form if it is in the form: Ax + By = C, where A, B, and C are integers and both A and B cannot be zero. We also saw that if the linear equation is solved for y, the constant term was the y-coordinate of the y-intercept. In other words, if y = mx + b, then (0, b) is the y-intercept. Now, let us find the slope of the line. The y- intercept, (0, b), is one point on the line. If we let x =, then y = m() + b = m + b. So, (, m + b) is a second point on the line. The slope is equal to y y x x = m+b b 0 = m = m. Thus, the coefficient of the x-term is m. Slope-intercept form A linear equation in two variables is said to be written in slope-intercept form if it is in the form y = mx + b where m is the slope of the line and the point (0, b) is the y-intercept. Given the graph below, find a) the slope, b) y-intercept, and c) the equation of the line: Ex. Ex.
a) The graph cross the a) The graph cross the at, so the y-intercept is at, so the y-intercept is (0, ). (0, ). b) Moving from the point b) Moving from the point (0, ) to (, 0), we have to (0, ) to (, ), we have to rise unit and run units. So, fall units and run unit. So, the slope is "rise" "run" =. c) Since m = "rise" the slope is "run" = =. and b =, c) Since m = and b =, the equation is y = x. the equation is y = x +. 05 Ex. Ex. a) The graph cross the a) The graph cross the at, so the y-intercept is at, so the y-intercept is (0, ). (0, ). b) Moving from the point b) Moving from the point (0, ) to (, 6), we have to (0, ) to (6, ), we have to fall units and run units. So, rise unit and run 6 units. So, the slope is "rise" "run" = = "rise". the slope is "run" = 6. c) Since m = and b =, c) Since m = 6 the equation is y = x. the equation is y = 6 x +. and b =, We can also go backwards to find the slope. In example #, if we move from (0, ) to (, ), we would rise units and run units backwards, So, our slope would be "rise" "run" = =. Likewise, in example #, if we went from (0, ) to ( 6, ), we would fall unit and run 6 units backwards. So, our slope would be "rise" "run" = 6 = 6.
06 Ex. 5 Ex. 6 a) The graph cross the a) The graph does not cross the at, so the y-intercept is, so there is no y-intercept. (0, ). b) This is a horizontal line so, b) This is a vertical line so, m = 0. m is undefined. c) The equation of a horizontal c) The equation of a vertical line line is in the form y = #. Since is in the form x = #. Since the line the line crosses the at, crosses the at, then the then the equation is y =. equation is x =. Find the slope and y-intercept and sketch the graph: Ex. 7 x + y = 5 First solve the equation for y: x + y = 5 + x = + x y = x 5 The slope is = = "rise" "run" and the y-intercept is (0, 5). Plot the point (0, 5). Then from that point rise units and run unit to get another point. From that new point, rise another units and run more unit to get the third point. Now, draw the graph. 5 y - axis - 5 - - - - 5 - x - axis - - - - 5
07 Ex. 8 x 6y = First solve the equation for y: x 6y = x = x 6y = x + 6 6 y = x The slope is = "rise" "run" and the y-intercept is (0, ). Plot the point (0, ). Then from that point rise unit and run units to get another point. From that new point, rise another unit and run more units to get the third point. Now, draw the graph. y - axis 5-5 - - - - 5 - x - axis - - - - 5 Ex. 9 8x + y = First solve the equation for y: 8x + y = 8x = 8x y = 8x + y = 8 x + 8 The slope is 8 = 8 and the y-intercept is (0, 8). Plot the point (0, 8). Then from that point fall 8 units and run units to get another point. From that new point fall another 8 units and run more units to get the third point. Now, draw the graph. Concept # = "rise" "run" y - axis -0-8 - 6 - - 6 8 0 - x - axis - Determining Whether Two Lines are Parallel, Perpendicular, or Neither. 0 8 6-6 - 8-0
08 Determine if the lines are parallel, perpendicular, or neither. Ex. 0 8x 7y = 6 Ex. x = y 8y = 7x + x + y = 5 First, solve each equation First, solve each equation for y to find the slope. for y to find the slope. 8x 7y = 6 x = y 8x = 8x y = x 7y = 8x + 6 m = 7 7 x + y = 5 y = 8 7 x 6 7 x = x m = 8 7 y = x + 5 8y = 7x + m = 8 8 y = 7 8 x + 8 But, m m, so they are not m = 7 8 parallel. Also, m m = ( ) Since m m = 8 7 ( 7 8) = 6, so they are not =, the lines are perpendicular. There the lines perpendicular. are neither. Ex. 9 x 6 y =.x 0.9y =.8 First, solve each equation for y to find the slope. 8 ( 9 x ) 8 ( 6 y ) = 8() 0(.x) 0(0.9y) = 0(.8) y ) = 8() x 9y = 8 ( x ) ( x y = 8 x = x x = x 9y = x + 8 y = x + 8 9 9 y = x 6 y = x m = Since m = m, then the lines are parallel. m =
09 Concept # Writing the Equation of the Line Given Its Slope and y-intercept Use the given information to write the equation of the line: Ex. The slope of the line is and it passes through (0, ). 8 m = and (0, b) = (0, ) which implies b =. 8 Plugging into y = mx + b, we get: y = 8 x So, the equation is y = 8 x. Ex. The slope of the line is and it passes through (0, ) m = and (0, b) = (0, ) which implies b =. Plugging into y = mx + b, we get: y = x + So, the equation is y = x +. Ex. 5 The slope of the line is 0 and it passes through (, 5). m = 0, but we do not have a y-intercept. A line with a slope of 0 is a horizontal line. Recall that a horizontal is in the form of y = constant. Since it has to pass through the point (, 5), then y has to be 5. So, the equation is y = 5. Ex. 6 The slope of the line is undefined and it passes through (, 5). m is undefined and we do not have a y-intercept. A line with a slope that is undefined is a vertical line. Recall that a vertical is in the form of x = constant. Since it has to pass through the point (, 5), then x has to be. So, the equation is x =.