Question

The equation of Line 1: 2x-4y=16
The equation of Line 2: y=-3/4x-3
The equation of Line 3: y-5=-2(x+1)
Which statement is true?

Line 1 has the steepest slope
Line 3 has a decreased graph
Line 2 has the biggest y-intercept
Line 3 has the least steep slope
Line 1 has the biggest y-intercept
Line 2 has an increasing graph

Answer 1

To analyze the given lines and determine which statements are true, we need to find their slopes and y-intercepts.

1. **Line 1: \(2x - 4y = 16\)**
- Rearranging it to slope-intercept form (\(y = mx + b\)):
\[
-4y = -2x + 16 \implies y = \frac{1}{2}x - 4
\]
- Slope (\(m_1\)): \(\frac{1}{2}\)
- y-intercept (\(b_1\)): \(-4\)

2. **Line 2: \(y = -\frac{3}{4}x - 3\)**
- Slope (\(m_2\)): \(-\frac{3}{4}\)
- y-intercept (\(b_2\)): \(-3\)

3. **Line 3: \(y - 5 = -2(x + 1)\)**
- Distributing:
\[
y - 5 = -2x - 2 \implies y = -2x + 3
\]
- Slope (\(m_3\)): \(-2\)
- y-intercept (\(b_3\)): \(3\)

Now, we can compare the slopes and y-intercepts of each line:

- **Slopes**:
- Line 1: \(\frac{1}{2} \approx 0.5\)
- Line 2: \(-\frac{3}{4} \approx -0.75\)
- Line 3: \(-2\)

The steepest slope corresponds to Line 1 \((\frac{1}{2})\) since it is positive and greater than the negative slopes of Lines 2 and 3.

- **Y-Intercepts**:
- Line 1: \(-4\)
- Line 2: \(-3\)
- Line 3: \(3\)

The largest y-intercept is from Line 3, which has a y-intercept \(3\).

- **Graph Behavior**:
- Line 1: increasing graph (positive slope)
- Line 2: decreasing graph (negative slope)
- Line 3: decreasing graph (steeper negative slope)

Now we can evaluate the statements:

1. **Line 1 has the steepest slope** - **True**
2. **Line 3 has a decreased graph** - **True** (but that was potentially meant to say "decreasing")
3. **Line 2 has the biggest y-intercept** - **False** (Line 3 has the biggest y-intercept)
4. **Line 3 has the least steep slope** - **False** (Line 1 has the steepest, Line 2 is less steep than Line 3)
5. **Line 1 has the biggest y-intercept** - **False** (Line 3 has the biggest y-intercept)
6. **Line 2 has an increasing graph** - **False** (it decreases due to the negative slope)

In conclusion, the only two true statements are:
- Line 1 has the steepest slope.
- Line 3 has a decreased (decreasing) graph.