Slope-Intercept and Point-Slope Forms of a Linear Equation Chapter 7 Section 4 Slope-Intercept and Point-Slope Forms of a Linear Equation

Learning Objective Write a linear equation in slope-intercept form Graph a linear equation using the slope and y-intercept Use the slope-intercept form to determine the equation of a line. Use the point-slope form to determine the equation of a line Compare the three methods of graphing linear equation. Key Vocabulary: slope, y-intercept, slope-intercept form, point-slope form

Slope-Intercept Form Standard form of a linear equation in two variables ax + by = c Slope-intercept form of a linear equation y = mx + b where m is the slope, and (0, b) is the y-intercept of the line To write an equation in slope-intercept form we solve for y. Graph is always a straight line with a slope of m and y-intercept of (0, b)

Slope-Intercept Form Examples y = 3x – 4 Slope = 3, y-intercept (0, -4) Positive slope , rises from left to right y = -2x + 5 Slope = -2, y-intercept (0, 5) Negative slope, falls from left to right Slope = , y-intercept (0, ) Positive slope, rises from left to right

Slope-Intercept Form Example: Write the equation 2x + 4y = 8 in slope-intercept form. State the slope and y-intercept. Negative slope Falls from left to right Down 1 and to the right 2

Perpendicular or Parallel Lines Two non-vertical lines with the same slope and different y-intercepts are parallel lines. Two lines whose slopes are negative reciprocals of each other are perpendicular lines.

Perpendicular or Parallel Lines Example: Determine if the two lines are perpendicular, parallel, or neither. 3x + y = 5 2y = -6x + 9 Two lines are parallel when their slopes are the same and the y-intercepts are different. When they do not intersect.

Perpendicular or Parallel Lines Example: Determine if the two lines are perpendicular, parallel, or neither. 5x - 4y = 8 4x + 5y = 10 To determine if two lines are perpendicular multiply the slopes of the two lines together. If the product is -1 then the slopes are negative reciprocals, and the lines are perpendicular.

Graph a Linear Equation using Slope and y-intercept Method 1: Graph by plotting Method 2: Graph by x- and y-intercept x-intercept (x, 0) y-intercept (0, y) Method 3: Graph by Slope and y-intercept Form Solve the equation for y Determine the y-intercept point (0,b) If a positive slope we move up and to the right If a negative slope we move down and to the right

Graph a Linear Equation using Slope and y-intercept Write 2x + 3y = 6 in slope-intercept form; then graph. (-3,4) (0,2) (0,0) (3,0) Down and to the right if negative

Graph a Linear Equation using Slope and y-intercept Graph -2x + 5y = 10 using the slope and y-intercept. (5, 4) (0,2) (0,0) Up and to the right if positive

Use the Slope-Intercept Form Determine the Equation of Line 1 First: Take two points and use the slope formula to determine the slope Second: determine the y-intercept (0, b) from the line? y-intercept (0,-5) Third: write the formula y = mx + b b is where the line crosses the y axis y = 2x - 5 (3, 1) (0,0) (0,-5)

Point-Slope Form of a Linear Equation When we know the slope and a point on the line we can use Point-Slope form to determine the equation y – y1 = m(x – x1) Where m is the slope of the line and (x1, y1) is a point on the line.

Point-Slope Form of a Linear Equation Write an equation, in slope-intercept form, of a line that goes through the point (-1, 4) and has a slope of 3. m = 3 goes through points (-1, 4) y-intercept (0,7) Standard Form ax + by = c -3x + y = 7 3x – y = -7

Point-Slope Form of a Linear Equation Write an equation, in slope-intercept form, of a line that goes through the point (8, -2) and has a slope of m = goes through points (8,-2) y-intercept (0,-8) Standard Form ax + by = c x + y = -8 Multiply by -1 x – y = 8

Slope Intercept Form and Point-Slope Form Sometimes we may have to use both formulas to find the equation. Find an equation of the line through the points (-1, 5) and (3,-3) Write the equation in slope-intercept form. m = -2 goes through points (-1, 5) and (3, -3) y-intercept (0,3) Standard Form ax + by = c 2x + y = 3

Remember Positive and negative slopes. Positive Slopes move up x number of units and to the right x number of units. Negative Slopes move down x number of units and to the right x number of units.

Remember If the linear equation does not have a constant term, the y-intercept is the origin (0, 0) y = mx Standard Form ax + by = c Slope-Intercept Form y = mx + b Point-Slope Form y – y1 = m(x – x1) Slope

Remember Writing linear equations in slope-intercept form If you know the slope and y-intercept form start with slope-intercept form. y = mx + b If you know the slope and a point on the line start with point-slope form. y – y1 = m(x – x1) If you know two point on the line start by finding the slope then use the point-slope form. y – y1 = m(x – x1) to find the equation.

HOMEWORK 7.4 Page 463 – 464: #9, 11, 14, 17, 19, 39, 49, 57, 59, 63