Explore $$ \begin{array}{l}\text { The following lines are hidden in the graph: } \ y=3 x+6 \ y=3(x-5) \ y=-\frac{1}{2} x-15 \ -x+y=12\end{array} $$ Capture all of the lines by writing coordinates for points on the lines.

Question

UpStudy AI · Accepted Answer

Certainly! To capture each of the given lines, we'll identify several points that lie on each line by choosing specific $$ x $$-values and calculating the corresponding $$ y $$-values.

### 1. Line: $$ y = 3x + 6 $$

**Points:**
- When $$ x = 0 $$:
  $$
  y = 3(0) + 6 = 6 \quad \Rightarrow \quad (0, 6)
  $$
- When $$ x = 1 $$:
  $$
  y = 3(1) + 6 = 9 \quad \Rightarrow \quad (1, 9)
  $$
- When $$ x = -2 $$:
  $$
  y = 3(-2) + 6 = 0 \quad \Rightarrow \quad (-2, 0)
  $$

### 2. Line: $$ y = 3(x - 5) $$ or $$ y = 3x - 15 $$

**Points:**
- When $$ x = 5 $$:
  $$
  y = 3(5) - 15 = 0 \quad \Rightarrow \quad (5, 0)
  $$
- When $$ x = 0 $$:
  $$
  y = 3(0) - 15 = -15 \quad \Rightarrow \quad (0, -15)
  $$
- When $$ x = 10 $$:
  $$
  y = 3(10) - 15 = 15 \quad \Rightarrow \quad (10, 15)
  $$

### 3. Line: $$ y = -\frac{1}{2}x - 15 $$

**Points:**
- When $$ x = 0 $$:
  $$
  y = -\frac{1}{2}(0) - 15 = -15 \quad \Rightarrow \quad (0, -15)
  $$
- When $$ x = 2 $$:
  $$
  y = -\frac{1}{2}(2) - 15 = -1 - 15 = -16 \quad \Rightarrow \quad (2, -16)
  $$
- When $$ x = -4 $$:
  $$
  y = -\frac{1}{2}(-4) - 15 = 2 - 15 = -13 \quad \Rightarrow \quad (-4, -13)
  $$

### 4. Line: $$ -x + y = 12 $$ or $$ y = x + 12 $$

**Points:**
- When $$ x = 0 $$:
  $$
  y = 0 + 12 = 12 \quad \Rightarrow \quad (0, 12)
  $$
- When $$ x = -12 $$:
  $$
  y = -12 + 12 = 0 \quad \Rightarrow \quad (-12, 0)
  $$
- When $$ x = 4 $$:
  $$
  y = 4 + 12 = 16 \quad \Rightarrow \quad (4, 16)
  $$

---

## Summary of Points on Each Line

| **Line Equation**         | **Points**                     |
|---------------------------|--------------------------------|
| $$ y = 3x + 6 $$           | $$ (0, 6) $$, $$ (1, 9) $$, $$ (-2, 0) $$ |
| $$ y = 3x - 15 $$          | $$ (5, 0) $$, $$ (0, -15) $$, $$ (10, 15) $$ |
| $$ y = -\frac{1}{2}x - 15 $$| $$ (0, -15) $$, $$ (2, -16) $$, $$ (-4, -13) $$ |
| $$ y = x + 12 $$           | $$ (0, 12) $$, $$ (-12, 0) $$, $$ (4, 16) $$ |

You can plot these points on a graph to visualize each line. Connecting the points for each line will display all the hidden lines as described.
Here are points on each line: 1. **$$ y = 3x + 6 $$:** - $$ (0, 6) $$ - $$ (1, 9) $$ - $$ (-2, 0) $$ 2. **$$ y = 3x - 15 $$:** - $$ (5, 0) $$ - $$ (0, -15) $$ - $$ (10, 15) $$ 3. **$$ y = -\frac{1}{2}x - 15 $$:** - $$ (0, -15) $$ - $$ (2, -16) $$ - $$ (-4, -13) $$ 4. **$$ y = x + 12 $$:** - $$ (0, 12) $$ - $$ (-12, 0) $$ - $$ (4, 16) $$ Plot these points to visualize the lines.

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Capture all of the lines by writing coordinates for points on the
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Explore Capture all of the lines by writing coordinates for points on the lines.