Which of the following equations represents a line that is parallel to the line with equation
` y = -3x + 4 ` ?

Options

A. ` 6x + 2y = 15 `
B. ` 3x - y = 7 `
C. ` 2x - 3y = 6 `
D. ` x + 3y = 1 `

Explanation

To determine which of the given equations represents a line that is parallel to the line described by the equation \( y = -3x + 4 \), we need to follow these steps:

### Step 1: Identify the slope of the given line.
The equation \( y = -3x + 4 \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. From this equation, we can see that the slope \( m \) is \(-3\).

**Hint:** Remember that in the slope-intercept form, the coefficient of \( x \) represents the slope of the line.

### Step 2: Understand the condition for parallel lines.
Two lines are parallel if and only if they have the same slope. Therefore, we need to find an equation among the options that also has a slope of \(-3\).

**Hint:** Look for the slope in each option by converting the equations to slope-intercept form.

### Step 3: Analyze each option to find its slope.

#### Option 1: \( 6x + 2y = 15 \)
1. Rearranging the equation:
\[ 2y = -6x + 15 \]
2. Dividing by 2:
\[ y = -3x + \frac{15}{2} \]
Here, the slope \( m = -3 \).

#### Option 2: \( 3x - y = 7 \)
1. Rearranging the equation:
\[ -y = -3x + 7 \]
2. Multiplying by \(-1\):
\[ y = 3x - 7 \]
Here, the slope \( m = 3 \).

#### Option 3: \( 2x - 3y = 6 \)
1. Rearranging the equation:
\[ -3y = -2x + 6 \]
2. Dividing by \(-3\):
\[ y = \frac{2}{3}x - 2 \]
Here, the slope \( m = \frac{2}{3} \).

#### Option 4: \( x + 3y = 1 \)
1. Rearranging the equation:
\[ 3y = -x + 1 \]
2. Dividing by 3:
\[ y = -\frac{1}{3}x + \frac{1}{3} \]
Here, the slope \( m = -\frac{1}{3} \).

### Step 4: Compare slopes.
From our analysis:
- Option 1 has a slope of \(-3\) (parallel).
- Option 2 has a slope of \(3\) (not parallel).
- Option 3 has a slope of \(\frac{2}{3}\) (not parallel).
- Option 4 has a slope of \(-\frac{1}{3}\) (not parallel).

### Conclusion:
The only equation that has the same slope as the given line \( y = -3x + 4 \) is from **Option 1: \( 6x + 2y = 15 \)**.

**Final Answer:** Option 1 represents a line that is parallel to the line with equation \( y = -3x + 4 \).

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