Forms of Linear Equations

Forms of linear equations

There are three main forms of linear equations.

Homework will be on DeltaMath

Finish this presentation for classwork.

Standard Form

Ax+By=C, where A, B, and C are all integers (no fractions or decimals allowed!!) and A is positive.

Standard Form

Ax+By=C, where A, B, and C are all integers (no fractions or decimals allowed!!) and A is positive.

-3x+7y=19 is not standard form because A is negative.

4x - 0.5y = 3 is not standard form, because B is not an integer.

71x - 83y = -212,024 is in standard form. A, B, and C are all integers, and A is positive.

Multiple Choice

Rewrite the equation in standard form

$$y\ =\ \frac{1}{3}x\ -\frac{9}{2}$$  

• $$-\frac{1}{3}x\ +y=-\frac{9}{2}$$  

• $$-2x+6y=-27$$  

• $$2x-6y=27$$  

• $$2x+6y=-27$$  

Multiple Choice

Standard form is useful for quickly finding the x and y-intercepts.

What are the x and y-intercepts of the equation 2x - 6y = 27?

• $$x=\frac{27}{2},\ y=-\frac{9}{2}$$

• $$x=2,\ y=-6$$

• $$x=2,\ y=6$$

• $$x=-2,\ y=6$$

Standard Form

To find the x-intercept for the equation 2x - 6y = 27, substitute 0 into y, then divide 27 by 2.

To find the y-intercept, substitute 0 into x, then divide 27 by -6 and reduce.

Slope-Intercept Form

y = mx + b, where m is the slope (or rate of change) and b is the y-intercept (or start of function).

Slope-Intercept Form

y = mx + b, where m is the slope (or rate of change) and b is the y-intercept (or start of function).

-2y = 4x + 3 is not in slope-intercept form, because y has a coefficient.

x = -4 is not in slope-intercept form, because it is not a function. The slope is undefined (it's graph is a vertical line).

 $y\ =\ -\frac{4}{3}x-\frac{19}{8}$  is in slope-intercept form.  $-\frac{4}{3}$  is the slope, and  $-\frac{19}{8}$  is the y-intercept.

Write the equation of the line

Multiple Choice

What was the equation of the line on the previous slide?

• $$y=4x-1$$

• $$y=-4x+\frac{1}{4}$$

• $$y=-\frac{1}{4}x-4$$

• $$y=\frac{1}{4}x-4$$

Point-Slope Form

y-y₁ = m(x - x₁), where m is the slope and (x₁ , y₁) is a point on the line.

Point-Slope Form

y-y1 = m(x - x1), where m is the slope and (x1 , y1) is a point on the line.

 $y+4=\frac{1}{4}\left(x-3\right)$  is in point-slope form. The slope is  $\frac{1}{4}$  and it goes through the point (3, -4)

Multiple Choice

Write the equation in point-slope form for the line that goes through (-2, -8) and has a slope of 1.

• $$y-8=x-2$$

• $$y-2=x-8$$

• $$y+2=x+8$$

• $$y+8=x+2$$

Parallel and Perpendicular Lines

It's not that hard, really.

Parallel and Perpendicular Lines

Parallel lines have the same slope.
 $m=m$  

Perpendicular lines have slopes that are opposite reciprocals. 

(Change the sign and flip the fraction).

 $m=-\frac{1}{m}$  

Multiple Choice

Write the equation in point-slope form of the line parallel to

$$y=\frac{1}{2}x+3$$  passing through the point (-6,5)

• $$y-5=-2\left(x+6\right)$$  

• $$y=\frac{1}{2}x+8$$  

• $$y+5=\frac{1}{2}\left(x-6\right)$$  

• $$y-5=\frac{1}{2}\left(x+6\right)$$  

Get ready for some practice

Multiple Choice

Write an equation of a line that passes through the point (5,-1) and is parallel to the line y = (-3/5)x -3

• y = (-3/5)x + 2

• y = (-3/5)x -2

• y = (3/5)x + 2

• y = (5/3) + 2

Multiple Choice

Write an equation for a line perpendicular to
y = -5x + 3 through (-5, -4)

• y = -3x + 1/5
• y = -3x - 1/5
• y = 1/5x - 3
• y = -1/5x - 3

Multiple Choice

Are these lines perpendicular?

y = 4x + 2

y = (-1/4)x + 12

• YES

• NO

Multiple Choice

Are these lines parallel?

y = (-1/2)x + 1

y = (1/2)x - 5

• YES

• NO

Multiple Choice

What is the slope of a line perpendicular to -10/3

• -10/3

• 10/3

• 3/10

• -3/10

Multiple Choice

What slope is parallel to:  
m = 4

• 4
• -1/4
• -4
• 1/4

Multiple Choice

What slope is parallel to:  
m = 3/4

• 4/3
• 3/4
• -3/4
• -4/3

Multiple Choice

What is the equation of the line?

• y=3x
• y=3
• x=3
• y=-3

Multiple Choice

Are these two lines parallel, perpendicular or neither?
y = -1/3x - 5
y = 1/3x + 2

• Parallel
• Perpendicular
• Neither

Multiple Choice

Write the equation of a line PARALLEL to 3x - y = 2 that passes through the point (1, 5).

• y = 3x - 14

• y = 3x + 2

• $$y=\frac{1}{3}x+6$$

• $$y=\frac{1}{3}x-4$$

Multiple Choice

Write the equation of a line PERPENDICULAR to $$y=\frac{1}{3}x-9$$  that passes through the point (-6, 10).

• $$y=\frac{1}{3}x-9$$

• $$y=\frac{1}{3}x+10$$

• $$y=-3x+6$$

• $$y=-3x-8$$

Multiple Choice

Write the equation of a line PERPENDICULAR to y = -2x + 1 that passes through the point (2, 3).

• $$y=-\frac{1}{2}x+\frac{1}{2}$$

• $$y=\frac{1}{2}x+2$$

• $$y=-2x+7$$

• $$y=-2x+8$$

Multiple Choice

Write the equation of a line PARALLEL to y = 2x + 3 that passes through the point (3, 1).

• $$y=2x\ -3$$

• $$y=2x-5$$

• $$y=2x\ +\ 1$$

• $$y=-\frac{1}{2}x+\frac{5}{2}$$

Don't forget: Homework is on DeltaMath

Parallel and Perpendicular Lines