Chapter 3: Graphing

3.4 Graphing Linear Equations

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form y = mx + b, then m gives the rise over run value and the value b gives the point where the line crosses the y-axis, also known as the y-intercept.

Example 3.4.1

Given the following equations, identify the slope and the y-intercept.

  1. \begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}
  2. \begin{array}{lll} y = \dfrac{1}{2}x - 1\hspace{0.08in} & \text{Slope }(m)=\dfrac{1}{2}\hspace{0.1in}&y\text{-intercept } (b)=-1 \end{array}
  3. \begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}
  4. \begin{array}{lll} y = \dfrac{2}{3}x\hspace{0.34in} & \text{Slope }(m)=\dfrac{2}{3}\hspace{0.1in} &y\text{-intercept } (b)=0 \end{array}

When graphing a linear equation using the slope-intercept method, start by using the value given for the y-intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

Example 3.4.2

Graph the equation y = 2x - 3.

First, place a dot on the y-intercept, y = -3, which is placed on the coordinate (0, -3).

image

Now, place the next dot using the slope of 2.

A slope of 2 means that the line rises 2 for every 1 across.

Simply, m = 2 is the same as m = \dfrac{2}{1}, where \Delta y = 2 and \Delta x = 1.

Placing these points on the graph becomes a simple counting exercise, which is done as follows:

For m = 2, go up 2 and forward 1 from each point.
image

Once several dots have been drawn, draw a line through them, like so:

image

Note that dots can also be drawn in the reverse of what has been drawn here.

Slope is 2 when rise over run is \dfrac{2}{1} or \dfrac{-2}{-1}, which would be drawn as follows:

For m = 2, go down 2 and back 1 from each point.

 

Example 3.4.3

Graph the equation y = \dfrac{2}{3}x.

First, place a dot on the y-intercept, (0, 0).

Now, place the dots according to the slope, \dfrac{2}{3}.

When m = 2 over 3, go up 2 and forward 3 to get the next point.

This will generate the following set of dots on the graph. All that remains is to draw a line through the dots.

Line with slope 2 over 3. Passes through (−3, −2), (0, 0), (3, 2), and (6, 4).

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find x when y = 0 and find y when x = 0.

Example 3.4.4

Graph the equation 2x + y = 6.

To find the first coordinate, choose x = 0.

This yields:

    \[\begin{array}{lllll} 2(0)&+&y&=&6 \\ &&y&=&6 \end{array}\]

Coordinate is (0, 6).

Now choose y = 0.

This yields:

    \[\begin{array}{llrll} 2x&+&0&=&6 \\ &&2x&=&6 \\ &&x&=&\frac{6}{2} \text{ or } 3 \end{array}\]

Coordinate is (3, 0).

Draw these coordinates on the graph and draw a line through them.

image

Example 3.4.5

Graph the equation x + 2y = 4.

To find the first coordinate, choose x = 0.

This yields:

    \[\begin{array}{llrll} (0)&+&2y&=&4 \\ &&y&=&\frac{4}{2} \text{ or } 2 \end{array}\]

Coordinate is (0, 2).

Now choose y = 0.

This yields:

    \[\begin{array}{llrll} x&+&2(0)&=&4 \\ &&x&=&4 \end{array}\]

Coordinate is (4, 0).

Draw these coordinates on the graph and draw a line through them.

image

Example 3.4.6

Graph the equation 2x + y = 0.

To find the first coordinate, choose x = 0.

This yields:

    \[\begin{array}{llrll} 2(0)&+&y&=&0 \\ &&y&=&0 \end{array}\]

Coordinate is (0, 0).

Since the intercept is (0, 0), finding the other intercept yields the same coordinate. In this case, choose any value of convenience.

Choose x = 2.

This yields:

    \[\begin{array}{rlrlr} 2(2)&+&y&=&0 \\ 4&+&y&=&0 \\ -4&&&&-4 \\ \midrule &&y&=&-4 \end{array}\]

Coordinate is (2, -4).

Draw these coordinates on the graph and draw a line through them.

image

Questions

For questions 1 to 10, sketch each linear equation using the slope-intercept method.

  1. y = -\dfrac{1}{4}x - 3
  2. y = \dfrac{3}{2}x - 1
  3. y = -\dfrac{5}{4}x - 4
  4. y = -\dfrac{3}{5}x + 1
  5. y = -\dfrac{4}{3}x + 2
  6. y = \dfrac{5}{3}x + 4
  7. y = \dfrac{3}{2}x - 5
  8. y = -\dfrac{2}{3}x - 2
  9. y = -\dfrac{4}{5}x - 3
  10. y = \dfrac{1}{2}x

For questions 11 to 20, sketch each linear equation using the x\text{-} and y-intercepts.

  1. x + 4y = -4
  2. 2x - y = 2
  3. 2x + y = 4
  4. 3x + 4y = 12
  5. 2x - y = 2
  6. 4x + 3y = -12
  7. x + y = -5
  8. 3x + 2y = 6
  9. x - y = -2
  10. 4x - y = -4

For questions 21 to 28, sketch each linear equation using any method.

  1. y = -\dfrac{1}{2}x + 3
  2. y = 2x - 1
  3. y = -\dfrac{5}{4}x
  4. y = -3x + 2
  5. y = -\dfrac{3}{2}x + 1
  6. y = \dfrac{1}{3}x - 3
  7. y = \dfrac{3}{2}x + 2
  8. y = 2x - 2

For questions 29 to 40, reduce and sketch each linear equation using any method.

  1. y + 3 = -\dfrac{4}{5}x + 3
  2. y - 4 = \dfrac{1}{2}x
  3. x + 5y = -3 + 2y
  4. 3x - y = 4 + x - 2y
  5. 4x + 3y = 5 (x + y)
  6. 3x + 4y = 12 - 2y
  7. 2x - y = 2 - y \text{ (tricky)}
  8. 7x + 3y = 2(2x + 2y) + 6
  9. x + y = -2x + 3
  10. 3x + 4y = 3y + 6
  11. 2(x + y) = -3(x + y) + 5
  12. 9x - y = 4x + 5

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-3-4/”>Answer Key 3.4

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