Graphing Linear Equations Worksheets with Answer Key - Free Printable

Here are the step-by-step instructions to graph each problem on your worksheet.

1) $y > -x - 5$

* Identify the line: The equation is in slope-intercept form ($y = mx + b$). The y-intercept ($b$) is -5, and the slope ($m$) is -1.
* Draw the boundary: Start at $(0, -5)$ on the y-axis. Since the slope is $-1$, go down 1 unit and right 1 unit to plot another point (like $(1, -6)$). Connect these points with a dashed line because the symbol is $>$ (strictly greater than).
* Shade the region: Pick a test point like $(0,0)$. Is $0 > -0 - 5$? Yes, $0 > -5$ is true. So, shade the area above the line (the side containing the origin).

2) $y = -3x - 3$

* Identify the line: The y-intercept ($b$) is -3, and the slope ($m$) is -3.
* Draw the line: Start at $(0, -3)$ on the y-axis. Since the slope is $-3$ (or $\frac{-3}{1}$), go down 3 units and right 1 unit to get to $(1, -6)$. You can also go up 3 and left 1 to get to $(-1, 0)$. Draw a solid line through these points.
* Shading: There is no inequality symbol, so do not shade any region. Just draw the line.

3) $x < -5$

* Identify the line: This is a vertical line where the x-value is always -5.
* Draw the boundary: Find $-5$ on the horizontal x-axis. Draw a straight vertical line up and down through $x = -5$. Make this line dashed because the symbol is $<$ (strictly less than).
* Shade the region: We want x-values *less than* -5. These numbers are to the left of -5 on the number line. Shade everything to the left of the dashed line.

4) $x - 4y = 0$

* Identify the line: Let's find two points to plot.
* If $x = 0$, then $0 - 4y = 0$, so $y = 0$. Point: $(0,0)$.
* If $x = 4$, then $4 - 4y = 0 \rightarrow 4 = 4y \rightarrow y = 1$. Point: $(4,1)$.
* Alternatively, rewrite as $y = \frac{1}{4}x$. Slope is $\frac{1}{4}$, y-intercept is $0$.
* Draw the line: Plot $(0,0)$ and $(4,1)$. Draw a solid line connecting them.
* Shading: No inequality symbol, so do not shade.

5) $5x - 3y \leq -15$

* Find intercepts: It is easiest to graph standard form equations using intercepts.
* x-intercept (set $y=0$): $5x = -15 \rightarrow x = -3$. Plot $(-3, 0)$.
* y-intercept (set $x=0$): $-3y = -15 \rightarrow y = 5$. Plot $(0, 5)$.
* Draw the boundary: Connect $(-3, 0)$ and $(0, 5)$ with a solid line because the symbol is $\leq$ (includes "equal to").
* Shade the region: Test the point $(0,0)$. Is $5(0) - 3(0) \leq -15$? Is $0 \leq -15$? No, that is false. Therefore, shade the side not containing $(0,0)$. Shade below/left of the line.

6) $y = 3x - 4$

* Identify the line: The y-intercept ($b$) is -4, and the slope ($m$) is 3.
* Draw the line: Start at $(0, -4)$ on the y-axis. Since the slope is $3$ (or $\frac{3}{1}$), go up 3 units and right 1 unit to get to $(1, -1)$. Draw a solid line through these points.
* Shading: No inequality symbol, so do not shade.

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Final Answer:
Graphing Instructions Summary:

1. $y > -x - 5$: Dashed line passing through $(0, -5)$ and $(1, -6)$. Shade above the line.
2. $y = -3x - 3$: Solid line passing through $(0, -3)$ and $(1, -6)$. No shading.
3. $x < -5$: Dashed vertical line at $x = -5$. Shade to the left.
4. $x - 4y = 0$: Solid line passing through $(0, 0)$ and $(4, 1)$. No shading.
5. $5x - 3y \leq -15$: Solid line passing through $(-3, 0)$ and $(0, 5)$. Shade below/left (away from origin).
6. $y = 3x - 4$: Solid line passing through $(0, -4)$ and $(1, -1)$. No shading.