Linear Equations in Various Forms

Linear Equations in Various Forms

Today we will explore various forms of equations for linear functions in two variables.

Multiple Choice

A student ran a cross country race from school to a local park. The graph shows the student’s distance in miles from the park after running for x minutes.

What is the domain of the function for this situation?

• All real numbers less than or equal to 40.

• All real numbers less than or equal to 7.

• All real numbers greater than or equal to 0 and less than or equal to 40.

• All real numbers greater than or equal to 0 and less than or equal to 7.

Multiple Choice

What are the equation and slope of the line shown on the coordinate grid?

• $y=3$ ; slope is zero

• $x=3$ ; slope is $-\frac{1}{3}$

• $y=3$ ; slope is $3$

• $x=3$ ; slope is undefined

Poll

How do you feel about your ability to determine and interpret the slope of a function?

• Totally got this. Easy dubs.

• Kinda got this. It makes a little sense.

• Eh, sometimes I get it. Depends on who is sitting next to me.

• 10/10 do not recommend letting me do this alone.

Slope-Intercept Form

 $y=mx+b$  

• m is the slope

• b is the y-coordinate of the y-intercept

Multiple Choice

Given $y=3x+4$  , the slope of this linear function is

•  $m=3$  

•  $m=4$  

•  $m=\frac{3}{4}$  

•  $m=\frac{4}{3}$  

Multiple Choice

Given $y=3x+4$  , the y-intercept of this linear function is

•  $\left(0,3\right)$  

•  $\left(3,0\right)$  

•  $\left(0,4\right)$  

•  $\left(4,0\right)$  

Slope-Intercept Form

Take note that while the proper form of the equation is written  $y=mx+b$  , you can move the terms around. The slope is always the coefficient of the x and the constant is always the y-coordinate of the y-intercept.

Multiple Select

Given  $y=-2+\frac{3}{4}x$  , identify the slope and the y-intercept. 

• The slope of the linear function is  $\frac{3}{4}$  because  $\frac{3}{4}$  is the coefficient of the independent variable.

• The slope of the linear function is  $-2$  because  $-2$  is the coefficient of the independent variable.

• The y-intercept of the linear function is  $\left(0,-2\right)$  because  $-2$  is the constant in the equation.

• The y-intercept of the linear function is  $\left(0,\frac{3}{4}\right)$  because  $\left(0,\frac{3}{4}\right)$  is the constant in the equation.

Multiple Select

Given  $y=-\frac{2}{3}+3x$  , identify the slope and the y-intercept. 

• The slope of the linear function is  $-\frac{2}{3}$  because  $-\frac{2}{3}$  is the coefficient of the independent variable.

• The slope of the linear function is  $3$  because  $3$  is the coefficient of the independent variable.

• The y-intercept of the linear function is  $\left(0,-\frac{2}{3}\right)$  because  $\left(0,-\frac{2}{3}\right)$  is the constant in the equation.

• The y-intercept of the linear function is  $\left(0,3\right)$  because  $\left(0,3\right)$  is the constant in the equation.

Slope-Intercept Form

When given a graph, you can determine the equation of a linear function by:

• Finding the y-intercept and another second point on the line.
• Calculating the slope of the line.
• Replacing those values into the equation.

Multiple Choice

Given the graphed function, determine the equation of the line in slope-intercept form.

• $y=-\frac{2}{5}x-1$

• $y=\frac{5}{2}x-1$

• $y=\frac{2}{5}x-1$

• $y=\frac{2}{5}x+1$

Multiple Choice

Given the table of values, determine the equation of the line in slope-intercept form.

•  $y=\frac{1}{1.3}x+2$ 

•  $y=-\frac{1}{1.3}x+2$ 

•  $y=1.3x+2$ 
  

•  $y=-1.3x+2$  

Multiple Choice

Given that equation has a slope of  $\sqrt{\frac{2}{3}}$  and a y-intercept of  $\left(0,\frac{\pi}{3}\right)$  , determine the equation of the line in slope-intercept form.

•  $y=\sqrt{\frac{2}{3}}x-\frac{\pi}{3}$ 

•  $y=\sqrt{\frac{2}{3}}x+\frac{\pi}{3}$ 

•  $y=\sqrt{\frac{\pi}{3}}x+\frac{2}{3}$ 
  

•  $y=\sqrt{\frac{\pi}{3}}x+\frac{2}{3}$  

Point-Slope Form

 $y-y_1=m\left(x-x_1\right)$  

• m is the slope

•  $\left(x_1,y_1\right)$  is a given point

Multiple Choice

Given that the line of an equation contains the point  $\left(2,3\right)$  and has a slope of  $\frac{1}{3}$  , determine the equation of the line in point-slope form.

•  $y-2=\frac{1}{3}\left(x-3\right)$ 

•  $y-3=\frac{1}{3}\left(x-2\right)$ 

•  $y+2=\frac{1}{3}x\left(x+3\right)$ 
  

•  $y+3=\frac{1}{3}\left(x+2\right)$  

Multiple Choice

Given that the line of an equation contains the point  $\left(-3,2\right)$  and has a slope of  $\frac{1}{6}$  , determine the equation of the line in point-slope form.

•  $y-3=\frac{1}{6}\left(x-2\right)$ 

•  $y-2=\frac{1}{6}\left(x-3\right)$ 

•  $y-2=\frac{1}{6}\left(x+3\right)$ 
  

•  $y+3=\frac{1}{6}\left(x+2\right)$  

Standard Form

 $Ax+By=C$  

• x and y terms are on one side of the equation and the constant (C) is on the other side

•  $-\frac{A}{B}$  is the slope

Multiple Choice

Given the equation $2x+3y=9$ , what is the slope of the linear equation?

•  $-\frac{2}{3}$  

•  $-\frac{3}{2}$  

•  $\frac{2}{3}$  

•  $\frac{3}{2}$  

Multiple Choice

Given the equation $-3x+4y=8$ , what is the slope of the linear equation?

•  $-\frac{3}{4}$  

•  $-\frac{4}{3}$  

•  $\frac{3}{4}$  

•  $\frac{4}{3}$  

Multiple Choice

Will’s meal of grilled chicken strips and carrots contained a total of 333 calories. The bag of grilled chicken strips contained x servings of 110 calories each. The bag of carrots contained y servings of 29 calories each. Which equation written in standard form represents the number of servings of chicken strips and carrots that Will may have eaten?

• $110y=-29x+333$

• $29x+110y=333$

• $29x=-110x+333$

• $110x+29y=333$

Practice Round

Try out these problems and see how you do!

Multiple Choice

The table represents some points on the graph of a linear function.

Which function represents the same relationship?

• $m\left(x\right)=\frac{1}{5}x-12.5$

• $m\left(x\right)=-5x+2.5$

• $m\left(x\right)=-\frac{1}{5}x+2.5$

• $m\left(x\right)=5x-12.5$

Multiple Choice

A function is graphed on the coordinate plane below.

Which linear equation in two variables represents the same relationship?

• $y=\frac{5}{2}x-3$

• $y=\frac{2}{5}x+1$

• $y=\frac{2}{5}x-3$

• $5x-2y=-6$

Multiple Choice

A coach recorded the time elapsed and distance completed by his Leg 2 runner on the relay marathon team. The table represents some points on the graph of a linear function.

Which equation represents the same relationship?

• $y-1\frac{3}{4}=\frac{2}{13}\left(x-\frac{1}{2}\right)$

• $y-1=\frac{13}{2}\left(x-5\right)$

• $y-3\frac{3}{8}=\frac{13}{2}\left(x-\frac{3}{4}\right)$

• $y-\frac{1}{8}=\frac{2}{13}\left(x-\frac{1}{4}\right)$

Multiple Select

Which of the equations below has a slope of  $\frac{1}{4}$  ?

•  $y=\frac{1}{4}x-5$  

•  $y-3\frac{1}{2}=\frac{1}{4}\left(x-2\frac{3}{4}\right)$  

•  $5x-20y=32$  

•  $5x+2y=32$  

Multiple Select

Which representations have the same rate of change as the equation $3x+5y=15$  ?

•  $y=\frac{3}{5}x+2$  

•  $y+7=-\frac{3}{5}\left(x-4\right)$