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.

Explore

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

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

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