Finding Linear Equations

Slope intercept form also allows us to easily find the equation of the line given the graph.  If you are given the slope and any point on the line you can find its equation by following the steps below:

Example: Find the equation of the line passing through (2, 3) with slope m = 1/2.

   Step 1: Use y = mx + b and find m.

   Step 2: Use the given point to find b.

   Step 3: Put it all together using y = mx + b.

Alternatively, you can use point-slope form to answer the same question.

Substitute (2, 3) and slope m = 1/2  into the formula as follows:

Either method is valid and the answer is the equation.

Find the equation of the given line.

Video Examples on YouTube:

Graph using Intercepts

You might be familiar with the basic fact that two points determine a line.  This fact leads to a nice and easy way to graph lines using the two points called the x- and y-intercepts.

All x-intercepts, if they exist, must have a corresponding y-value of zero.  All y-intercepts must have a corresponding x-value of zero.  This might sound confusing but just remember the following steps to algebraically find intercepts.

Example: Graph 3x − 5y = 15 using the x- and y-intercepts.

Plot the points and draw a line through them with a straight edge.

Instructional Video: Graphing by Finding Intercepts

This is a nice and easy method for determining the two points you need for graphing a line.  In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.  Be careful not to say that y = −3 is the y-intercept because the intercepts, actually, are ordered pairs or points on the graph so you should take care to say (0,−3) is the y-intercept.

Use the given graph to answer the question.

  

Be sure to pay attention to the scale. Misreading the scale is the most common error in this type of problem.
  

Example: Graph −4x + 3y = 12 using the intercepts.

             

Example: Graph −4x + 2y = −6 using the intercepts.

              

Example: Graph y = −5x +15 using the intercepts.

               

Example: Graph y = −3/4 x + 9 using the intercepts.

                 

This brings us to one of the most popular questions in linear graphing.  Do all lines have x- and y-intercepts?  The answer is NO.  Horizontal lines, of the form y = b, do not necessarily have x-intercepts.  Vertical lines, of the form x = a, do not necessarily have y-intercepts.

Example: Graph y = 3.

           

Example: Graph x = −2.

        

Instructional Video: Graphing Horizontal and Vertical Lines

Video Examples on YouTube:  Graphing Linear Functions Playlist









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Interactive: Slope-Intercept Form

Interactive Instructions: Slope-intercept form: y = mx + b. Move the green points to change m and b.

[ NOTES: Slope and y-Intercept ]

Dynamic Quiz - Solving Basic Linear Equations

Test your skills and answer 5 questions correctly.

Try this! Your score: 0/5

Solve for the given variable.

\(2x+3=13\)

5

\(x=\)

Push the Check button for feedback.

\(x=5\)

\(2x+3=13\)

Add \(-3\) to both sides.

\(2x=10\)

Divide both sides by \(2\).

\(x=5\)

Chapter 2 Sample Test Questions

Click here for a worksheet containing 20 sample test questions with answers.

[ Sample Test Questions Chapter 2 ]

Odd Integer Problem: Find three consecutive odd integers whose sum is 51. (Set up an algebraic equation then solve it.)

Perimeter Problem: The perimeter of a rectangle is 110 feet.  Find the dimensions if the length is 5 feet less than twice the width.