Which equation represents a line which is perpendicular to the line 6, x, plus, 7, y, equals, minus, 496*x*+7*y*=−49?

To find an equation that represents a line perpendicular to the given line \(6x + 7y = -49\), we need to follow these steps:

### Step 1: Find the slope of the given line
The equation of the line is \(6x + 7y = -49\). We need to rewrite this in the slope-intercept form \(y = mx + b\) to find the slope.

Start by solving for \(y\):
\[
6x + 7y = -49
\]
\[
7y = -6x - 49
\]
\[
y = -\frac{6}{7}x - \frac{49}{7}
\]
\[
y = -\frac{6}{7}x - 7
\]
The slope \(m_1\) of the given line is \(-\frac{6}{7}\).

### Step 2: Find the slope of the perpendicular line
The slope \(m_2\) of the line perpendicular to the given line is the negative reciprocal of \(m_1\). Thus:
\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{6}{7}} = \frac{7}{6}
\]

### Step 3: Use the slope to form the equation of the perpendicular line
The general form of the equation of a line in slope-intercept form is:
\[
y = mx + b
\]
Substituting \(m = \frac{7}{6}\), we have:
\[
y = \frac{7}{6}x + b
\]
This is the equation of any line that is perpendicular to the given line.

### Step 4: Choose a specific form if needed
If a specific equation is needed, you could use a point through which the line passes to determine \(b\). However, without a specific point, the general form of any line perpendicular to \(6x + 7y = -49\) is:
\[
\boxed{y = \frac{7}{6}x + b}
\]

Would you like to go over any details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you determine the slope of a line given in standard form?
2. What are the steps to convert an equation from slope-intercept form to standard form?
3. How do you find the equation of a line that is parallel to a given line?
4. What is the significance of the slope-intercept form in analyzing the steepness of a line?
5. How can you find the point of intersection between two perpendicular lines?

**Tip:** When finding the equation of a perpendicular line, always remember to take the negative reciprocal of the original slope to ensure the lines are perpendicular.

Learn how to find a line perpendicular to the equation 6x + 7y = -49 with a step-by-step solution. This problem covers geometric concepts such as slope and perpendicular lines. Suitable for high school students studying geometry.

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