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 [latex]y = mx + b[/latex], then [latex]m[/latex] gives the rise over run value and the value [latex]b[/latex] gives the point where the line crosses the [latex]y[/latex]-axis, also known as the [latex]y[/latex]-intercept.

Example 3.4.1

Given the following equations, identify the slope and the [latex]y[/latex]-intercept.

  1. [latex]\begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}[/latex]
  2. [latex]\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}[/latex]
  3. [latex]\begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}[/latex]
  4. [latex]\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}[/latex]

When graphing a linear equation using the slope-intercept method, start by using the value given for the [latex]y[/latex]-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 [latex]y = 2x - 3[/latex].

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

Graph with intercept at (0,-3)

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, [latex]m = 2[/latex] is the same as [latex]m = \dfrac{2}{1}[/latex], where [latex]\Delta y = 2[/latex] and [latex]\Delta x = 1[/latex].

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.

From each point, go up 2 and forward 1 to find the next point.

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

Line with slope of 2. Passes through (−3, 0), (1, −1), (2, 1), and (3, 3).

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

Slope is 2 when rise over run is [latex]\dfrac{2}{1}[/latex] or [latex]\dfrac{-2}{-1}[/latex], 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 [latex]y = \dfrac{2}{3}x[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex](0, 0)[/latex].

Now, place the dots according to the slope, [latex]\dfrac{2}{3}[/latex].

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 [latex]x[/latex] when [latex]y = 0[/latex] and find [latex]y[/latex] when [latex]x = 0[/latex].

Example 3.4.4

Graph the equation [latex]2x + y = 6[/latex].

To find the first coordinate, choose [latex]x = 0[/latex].

This yields:

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

Coordinate is [latex](0, 6)[/latex].

Now choose [latex]y = 0[/latex].

This yields:

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

Coordinate is [latex](3, 0)[/latex].

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

Line with slope of 2. Passes through (0, 6) and (3, 0).

Example 3.4.5

Graph the equation [latex]x + 2y = 4[/latex].

To find the first coordinate, choose [latex]x = 0[/latex].

This yields:

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

Coordinate is [latex](0, 2)[/latex].

Now choose [latex]y = 0[/latex].

This yields:

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

Coordinate is [latex](4, 0)[/latex].

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

Line with negative slope. Passes through (0, 2) and (4, 0).

Example 3.4.6

Graph the equation [latex]2x + y = 0[/latex].

To find the first coordinate, choose [latex]x = 0[/latex].

This yields:

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

Coordinate is [latex](0, 0)[/latex].

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

Choose [latex]x = 2[/latex].

This yields:

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

Coordinate is [latex](2, -4)[/latex].

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

Line with negative slope. Passes through (0, 0) and (2, −4).

Questions

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

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

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

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

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

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

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

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

Answer Key 3.4

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