Essential Question: How can you use the equations of two non-vertical lines to tell whether the equations are parallel or perpendicular?

Graphing a Linear EquationAny equation with both x and y can be graphed.A SOLUTION OF A LINEAR EQUATION IS ANY ORDERED PAIR (x, y) THAT MAKES THE EQUATION TRUE.In the equation y = 3x + 2 if x = 4 then y = 3(4) + 2 = 14Which means (4, 14) is a solution to that equation.There are infinite solutions to a linear equation, you just need to decide on an x value to start with.BECAUSE y DEPENDS ON THE VALUE OF x, WE CALL y THE DEPENDENT VARIABLE, AND CALL x THE INDEPENDENT VARIABLE

Graphing a Linear EquationYou can graph a linear equation by taking two solutions, putting them on a coordinate plane, and connecting a line through them.Example: Graph the equation y = 2/3x + 3

Graphing a Linear EquationYou can graph a linear equation by taking two solutions, putting them on a coordinate plane, and connecting a line through them.Example: Graph the equation y = 2/3x + 3Let x = 3, then y = 2/3(3) + 3 = 5Use the point (3, 5)

Graphing a Linear EquationYou can graph a linear equation by taking two solutions, putting them on a coordinate plane, and connecting a line through them.Example: Graph the equation y = 2/3x + 3Let x = 3, then y = 2/3(3) + 3 = 5Use the point (3, 5)Let x = -3, then y = 2/3(-3) + 3 = 1Use the point (-3,1)

Graphing a Linear EquationUse the point (3, 5)Use the point (-3,1) Connect the line

The y-intercept is where the graph crosses the y-axis.It can be found by setting x = 0 in a linear equation.The x-intercept is where the graph crosses the x-axis.It can be found by setting y = 0 in a linear equation.The intercepts can also be used in graphing a linear equation.

Example: The equation 3x + 2y = 120 models the number of passengers who can sit in a train car, where x is the number of adults and y is the number of children. Graph the equation. Explain what the x- and y-intercepts represent. Describe the domain and range.

3x + 2y = 120x-intercept3x + 2(0) = 1203x = 120x = 40

3x + 2y = 120x-intercept3x + 2(0) = 1203x = 120x = 40y-intercept3(0) + 2y = 1202y = 120y = 60

3x + 2y = 120x-intercept3x + 2(0) = 1203x = 120x = 40y-intercept3(0) + 2y = 1202y = 120y = 60Use the points (40,0) and (0,60)

SlopeSlope is found by taking the vertical change and dividing by the horizontal changeRise over RunThe formula is: where (x1,y1) and (x2,y2) represent solutions to a linear equation

SlopeExample: Find the slope of the line through the points (3,2) and (-9,6)

Writing Equations of LinesSlope-Intercept FormUsed when you know the slope and the y-intercepty = mx + bWhat does the m stand for? What does the b stand for?

Example: Find the slope of 4x + 3y = 7

Writing Equations of LinesSlope-Intercept FormUsed when you know the slope and the y-intercepty = mx + b

slope y-interceptOnce you get y by itself, the slope is the coefficient in front of the x.Example: Find the slope of 4x + 3y = 7

4x + 3y = 7Get y by itself

4x + 3y = 7 -4x -4x3y = -4x + 7

4x + 3y = 7 -4x -4x3y = -4x + 7 3 3 3y = -4/3 x + 7/3(leave as fractions)

4x + 3y = 7 -4x -4x3y = -4x + 7 3 3 3y = -4/3 x + 7/3(leave as fractions)

Slope = -4/3

4x + 3y = 7 -4x -4x3y = -4x + 7 3 3 3y = -4/3 x + 7/3(leave as fractions)

Slope = -4/3y-intercept = 7/3

AssignmentPage 671 19, 33 37 (odd problems)SHOW WORK

Essential Question: How can you use the equations of two non-vertical lines to tell whether the equations are parallel or perpendicular?

Writing Equations of LinesPoint-Slope FormUsed when you know a point on the line and the slopey y1 = m(x x1)Example: Write in slope-intercept form an equation of the line with slope -1/2 through the point (8,-1)

Slope = -1/2Point = (8, -1)y y1 = m(x x1)

Slope = -1/2Point = (8, -1)y y1 = m(x x1) (replace)y (-1) = -1/2(x 8)

Slope = -1/2Point = (8, -1)y y1 = m(x x1)(replace)y (-1) = -1/2(x 8)(distribute)y + 1 = -1/2x + 4

Slope = -1/2Point = (8, -1)y y1 = m(x x1) (replace)y (-1) = -1/2(x 8)(distribute)y + 1 = -1/2x + 4(subtract 1) -1-1y = -1/2 x + 3

Writing Equations of LinesSometimes, you will be given two points. In this case, you will first need to find the slope of the two points, then use either one of the points and the slope in point-slope formExample: Write in point-slope form an equation of the line through (1,5) and (4,-1)

Writing Equations of LinesSometimes, you will be given two points. In this case, you will first need to find the slope of the two points, then use either one of the points and the slope in point-slope formExample: Write in point-slope form an equation of the line through (1,5) and (4,-1)Find the slope:

Writing Equations of LinesExample: Write in point-slope form an equation of the line through (1,5) and (4,-1)Find the slope:

Choose either pointy y1 = m(x x1)y 5 = -2(x 1) OR y + 1 = -2(x 4) will give valid answers

The slopes of horizontal, vertical, parallel and perpendicular lines have special properties

Horizontal LineVertical LineParallel LinesPerpendicular Linesm = 0m = undefinedSlopes are equalSlopes are inverse reciprocalsy is constantx is constantFlip the fractionFlip the sign

Write an equation of the line through the point (6,1) and perpendicular to y = x + 2What is the slope of my starting line?

Write an equation of the line through the point (6,1) and perpendicular to y = x + 2What is the slope of my starting line? What is the slope of my perpendicular line?

Write an equation of the line through the point (6,1) and perpendicular to y = x + 2What is the slope of my starting line? What is the slope of my perpendicular line?Flip fraction = 4/3Flip sign = -4/3 Leave your answer in slope-intercept formYou have the slope, so find the interceptNote: You can also solve using point-slope form

Slope = -4/3Point = (6, 1)y = mx + bFind b1 = -4/3 (6) + b(replace)1 = -8 + b+8 +89 = by = -4/3 x + 9

AssignmentPage 68#21,23, 25 (Write in slope-intercept form)#27 31 (odd problems)#38 & 39