Linear Equations

What makes a linear equation LINEAR? An equation in one or more variables, each with an exponent of ONLY 1, where these variables are only added or subtracted.

So with that definition Which of these equations are linear? x+y = 5 2x+ 3y = 4 7x-3y = 14 y = 2x-2 y=4 x2 + y = 5 x = 5 xy = 5 x2 +y2 = 9 y = x2 y 3

So with that definition Which of these equations are linear? Not Linear x+y = 5 2x+ 3y = 4 7x-3y = 14 y = 2x-2 y=4 x2 + y = 5 x = 5 xy = 5 x2 +y2 = 9 y = x2 y 3

If you had to describe a line what characteristics would you detail? x y x Line A Line B

Slope, Intercepts y x y x Line A Line B

Slopes Positive Negative Horizontal Vertical

Intercepts – where the line crosses the axes. y x y x y-intercept=4 x-intercept=-3 x-intercept=-5 y-intercept=-5 Line A Line B

Intercepts are actually points in the coordinate system. x y x y-intercept=(0,4) x-intercept=(-3,0) x-intercept=(-5,0) y-intercept=(0,-5) Line A Line B

Quadrants Review y II I x III VI

Ordered Pairs Review : (x,y) II I (-x,y) (x,y) III VI (-x,-y) (x,-y)

Linear Equations – What you should be able to identify for all lines. The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope

Slope Intercept Standard Horizontal Vertical y = mx + b Ax + By = C Equation Forms Slope Intercept Standard Horizontal Vertical y = mx + b Ax + By = C y = b x = a

Slope-Intercept y = mx + b y = ½ x + 5 y = -3x - 7 Slope y-intercept

3x – 2y = 9 4x + 2y = 16 Standard Form Ax + By = C A, B, C are all integers with A > 0 3x – 2y = 9 4x + 2y = 16

Given our 4 example equations identify all of the following… The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope y = ½ x + 5 y = -3x – 7 3x – 2y = 9 4x + 2y = 16

y = ½ x + 5 Slope intercept The Equation Form Rising Direction ½ Slope -5/(½) = -10 -2 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope

y = -3x – 7 Slope intercept The Equation Form Falling Direction -3 -7 - -7/(-3) = -7/3 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope

3x – 2y = 9 Standard The Equation Form Rising Direction 3/2 Slope -2/3 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope

4x + 2y = 16 Standard The Equation Form Falling Direction -2 Slope 8 1/2 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope

What if you are just given two points on a line? The slope formula m = Similar to Point-Slope Form y – y1 = m(x – x1) or y2 – y1 = m(x2 – x1) y2 – y1 x2 – x1

1st – Find the Slope: y x A(6,6) B(-3,9)

15 9 5 3 slope = (6 - -9) (6 - -3) = = y x A(6,6) B(-3,9)

15 9 5 3 slope = (6 - -9) (6 - -3) = = y x A(6,6) B(-3,9)

Now substitute the slope and one point into the slope intercept form y = mx + b m = 5/3 point (6,6) 6 = (5/3)(6 + b) 6 = 10 + (5/3)b -4 = (5/3)b -12/5 = b Linear equation is y = (5/3)x – 12/3

31 Linear Equations On – Line Assignment