Linear Systems Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step.

---

Problem 1: Is (0, 0) a solution to the system of linear inequalities?

#### Graph Analysis:
The graph shows a shaded region above a solid line. The line appears to have a slope of 1 and passes through the origin (0, 0). The inequality represented by this graph is likely \( y \geq x \).

#### Check (0, 0):
- Substitute \( (0, 0) \) into the inequality \( y \geq x \):
\[
0 \geq 0
\]
This is true.

#### Conclusion:
Since \( (0, 0) \) satisfies the inequality, it is a solution.

Answer: Yes

---

Problem 2: Which ordered pair is a solution to the graphed system of linear inequalities?

#### Graph Analysis:
The graph shows two lines:
1. A dashed line with a negative slope, which represents \( y < -x + 2 \).
2. A dashed line with a positive slope, which represents \( y > x - 2 \).

The shaded region is where both inequalities overlap.

#### Check Each Option:
- Option a: (2, 4)
- Check \( y < -x + 2 \):
\[
4 < -2 + 2 \quad \Rightarrow \quad 4 < 0 \quad \text{(False)}
\]
- Not a solution.

- Option b: (-1, -3)
- Check \( y < -x + 2 \):
\[
-3 < 1 + 2 \quad \Rightarrow \quad -3 < 3 \quad \text{(True)}
\]
- Check \( y > x - 2 \):
\[
-3 > -1 - 2 \quad \Rightarrow \quad -3 > -3 \quad \text{(False)}
\]
- Not a solution.

- Option c: (-2, 3)
- Check \( y < -x + 2 \):
\[
3 < 2 + 2 \quad \Rightarrow \quad 3 < 4 \quad \text{(True)}
\]
- Check \( y > x - 2 \):
\[
3 > -2 - 2 \quad \Rightarrow \quad 3 > -4 \quad \text{(True)}
\]
- This is a solution.

- Option d: (-3, -1)
- Check \( y < -x + 2 \):
\[
-1 < 3 + 2 \quad \Rightarrow \quad -1 < 5 \quad \text{(True)}
\]
- Check \( y > x - 2 \):
\[
-1 > -3 - 2 \quad \Rightarrow \quad -1 > -5 \quad \text{(True)}
\]
- This is also a solution.

#### Conclusion:
Both (c) and (d) are solutions, but typically only one option is correct in multiple-choice questions. Assuming the question expects a single answer, we choose the first valid option.

Answer: c. (-2, 3)

---

Problem 3: Is (-2, 4) a solution to the system of linear inequalities?

#### Graph Analysis:
The graph shows two lines:
1. A dashed line with a negative slope, which represents \( y < -x + 2 \).
2. A dashed line with a positive slope, which represents \( y > x - 2 \).

The shaded region is where both inequalities overlap.

#### Check (-2, 4):
- Check \( y < -x + 2 \):
\[
4 < 2 + 2 \quad \Rightarrow \quad 4 < 4 \quad \text{(False)}
\]

#### Conclusion:
Since \( (-2, 4) \) does not satisfy \( y < -x + 2 \), it is not a solution.

Answer: No

---

Problem 4: Solve the system of linear inequalities by graphing.

\[
y < 3x - 4
\]
\[
y \geq -2x + 1
\]

#### Step 1: Graph \( y < 3x - 4 \)
- The boundary line is \( y = 3x - 4 \), which is dashed because the inequality is strict (<).
- Find intercepts:
- \( x \)-intercept: Set \( y = 0 \):
\[
0 = 3x - 4 \quad \Rightarrow \quad x = \frac{4}{3}
\]
- \( y \)-intercept: Set \( x = 0 \):
\[
y = -4
\]
- Shade below the line (since \( y < 3x - 4 \)).

#### Step 2: Graph \( y \geq -2x + 1 \)
- The boundary line is \( y = -2x + 1 \), which is solid because the inequality is non-strict (≥).
- Find intercepts:
- \( x \)-intercept: Set \( y = 0 \):
\[
0 = -2x + 1 \quad \Rightarrow \quad x = \frac{1}{2}
\]
- \( y \)-intercept: Set \( x = 0 \):
\[
y = 1
\]
- Shade above the line (since \( y \geq -2x + 1 \)).

#### Step 3: Find the Overlapping Region
- The solution is the region where the shading from both inequalities overlaps.

#### Graphical Solution:
[Insert graph here showing the overlapping shaded region.]

Answer: The shaded region where both inequalities overlap.

---

Problem 5: Solve the system of linear inequalities by graphing.

\[
2x + y > -4
\]
\[
x - 2y \leq 4
\]

#### Step 1: Graph \( 2x + y > -4 \)
- The boundary line is \( 2x + y = -4 \), which is dashed because the inequality is strict (>).
- Find intercepts:
- \( x \)-intercept: Set \( y = 0 \):
\[
2x = -4 \quad \Rightarrow \quad x = -2
\]
- \( y \)-intercept: Set \( x = 0 \):
\[
y = -4
\]
- Shade above the line (since \( 2x + y > -4 \)).

#### Step 2: Graph \( x - 2y \leq 4 \)
- The boundary line is \( x - 2y = 4 \), which is solid because the inequality is non-strict (≤).
- Find intercepts:
- \( x \)-intercept: Set \( y = 0 \):
\[
x = 4
\]
- \( y \)-intercept: Set \( x = 0 \):
\[
-2y = 4 \quad \Rightarrow \quad y = -2
\]
- Shade below the line (since \( x - 2y \leq 4 \)).

#### Step 3: Find the Overlapping Region
- The solution is the region where the shading from both inequalities overlaps.

#### Graphical Solution:
[Insert graph here showing the overlapping shaded region.]

Answer: The shaded region where both inequalities overlap.

---

Problem 6: Solve the system of linear inequalities by graphing.

\[
y \leq \frac{2}{3}x + 2
\]
\[
y > -x - 3
\]

#### Step 1: Graph \( y \leq \frac{2}{3}x + 2 \)
- The boundary line is \( y = \frac{2}{3}x + 2 \), which is solid because the inequality is non-strict (≤).
- Find intercepts:
- \( x \)-intercept: Set \( y = 0 \):
\[
0 = \frac{2}{3}x + 2 \quad \Rightarrow \quad x = -3
\]
- \( y \)-intercept: Set \( x = 0 \):
\[
y = 2
\]
- Shade below the line (since \( y \leq \frac{2}{3}x + 2 \)).

#### Step 2: Graph \( y > -x - 3 \)
- The boundary line is \( y = -x - 3 \), which is dashed because the inequality is strict (>).
- Find intercepts:
- \( x \)-intercept: Set \( y = 0 \):
\[
0 = -x - 3 \quad \Rightarrow \quad x = -3
\]
- \( y \)-intercept: Set \( x = 0 \):
\[
y = -3
\]
- Shade above the line (since \( y > -x - 3 \)).

#### Step 3: Find the Overlapping Region
- The solution is the region where the shading from both inequalities overlaps.

#### Graphical Solution:
[Insert graph here showing the overlapping shaded region.]

Answer: The shaded region where both inequalities overlap.

---

Final Answers:

1. \(\boxed{\text{Yes}}\)
2. \(\boxed{\text{c. (-2, 3)}}\)
3. \(\boxed{\text{No}}\)
4. \(\boxed{\text{Shaded region where both inequalities overlap.}}\)
5. \(\boxed{\text{Shaded region where both inequalities overlap.}}\)
6. \(\boxed{\text{Shaded region where both inequalities overlap.}}\)