Graphing Linear Equations Worksheets with Answer Key - Free Printable

Let’s solve each problem step by step. We’re graphing lines and inequalities on the coordinate plane. For each, we’ll find key points (like intercepts or slope) and then sketch accordingly — solid line for ≤ or ≥, dashed for < or >, and shade the correct side for inequalities.

---

Problem 1: y > -x - 5

This is an inequality. First, treat it like an equation to draw the boundary line:
y = -x - 5 → slope = -1, y-intercept = -5

Plot point (0, -5). From there, go down 1, right 1 → (1, -6), or up 1, left 1 → (-1, -4). Draw a dashed line because it’s “>” (not including the line).

Now, pick a test point not on the line — say (0,0). Plug into inequality:
0 > -0 - 5 → 0 > -5 → TRUE → so shade the side that includes (0,0) — which is ABOVE the line.

✔ Graph: Dashed line with slope -1 through (0,-5), shade above.

---

Problem 2: y = -3x - 3

This is a straight line equation. Slope = -3, y-intercept = -3.

Start at (0, -3). Go down 3, right 1 → (1, -6). Or up 3, left 1 → (-1, 0). Connect those points with a solid line (it’s an equality).

✔ Graph: Solid line through (0,-3) and (-1,0), steep downward slope.

---

Problem 3: x < -5

This is a vertical line inequality. The boundary is x = -5 (vertical line through x=-5).

Since it’s “<”, draw a dashed vertical line at x = -5.

Test point: Try (0,0). Is 0 < -5? No → FALSE → so do NOT shade the right side. Shade the LEFT side (where x is less than -5).

✔ Graph: Dashed vertical line at x = -5, shade everything to the left.

---

Problem 4: x - 4y = 0

Rewrite in slope-intercept form:
x - 4y = 0 → -4y = -x → y = (1/4)x

Slope = 1/4, y-intercept = 0 → passes through origin (0,0).

From (0,0), go up 1, right 4 → (4,1). Or down 1, left 4 → (-4,-1). Draw a solid line (equality).

✔ Graph: Solid line through (0,0) and (4,1), gentle upward slope.

---

Problem 5: 5x - 3y ≤ -15

First, rewrite in slope-intercept form:

5x - 3y ≤ -15
→ -3y ≤ -5x -15
→ Divide by -3 (remember to flip inequality sign!)
→ y ≥ (5/3)x + 5

So boundary line: y = (5/3)x + 5 → slope = 5/3, y-intercept = 5.

Plot (0,5). From there, go up 5, right 3 → (3,10) — but our grid only goes to y=8, so maybe go down 5, left 3 → (-3,0). That’s better.

Check: When x = -3, y = (5/3)(-3) + 5 = -5 + 5 = 0 → yes, (-3,0) is on the line.

Draw a solid line (because ≤ became ≥ after dividing by negative — still includes the line).

Now, test point: Use (0,0). Plug into original inequality:
5(0) - 3(0) ≤ -15 → 0 ≤ -15 → FALSE → so do NOT shade where (0,0) is. Shade the OTHER side — which is ABOVE the line (since y ≥ ...).

Wait — let’s double-check shading using the rewritten version: y ≥ (5/3)x + 5 → shade ABOVE the line. Yes.

But since (0,0) is below the line and gives false, we shade above — correct.

✔ Graph: Solid line through (0,5) and (-3,0), shade above the line.

---

Problem 6: y = 3x - 4

Straight line. Slope = 3, y-intercept = -4.

Start at (0, -4). Go up 3, right 1 → (1, -1). Or down 3, left 1 → (-1, -7). Draw solid line.

✔ Graph: Solid line through (0,-4) and (1,-1), steep upward slope.

---

Final Answer:

1. Dashed line y = -x - 5, shade above.
2. Solid line y = -3x - 3 through (0,-3) and (-1,0).
3. Dashed vertical line x = -5, shade left.
4. Solid line y = (1/4)x through (0,0) and (4,1).
5. Solid line y = (5/3)x + 5 through (0,5) and (-3,0), shade above.
6. Solid line y = 3x - 4 through (0,-4) and (1,-1).