Find the equation of the line that contains the point left parenthesis negative 3 comma negative 6 right parenthesis and is parallel to the line 3 x plus 4 y equals 7. Write the equation in​ slope-intercept form, if possible.

To find the equation of the line passing through the point \((-3, -6)\) and parallel to the line \(3x + 4y = 7\), we will follow these steps:

### Step 1: Find the slope of the given line
The given line is \(3x + 4y = 7\). To find the slope, we need to rewrite this in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope.

1. Start with the equation \(3x + 4y = 7\).
2. Solve for \(y\) by isolating it on one side:

\[
4y = -3x + 7
\]

\[
y = -\frac{3}{4}x + \frac{7}{4}
\]

So, the slope \(m\) of the given line is \(-\frac{3}{4}\).

### Step 2: Use the slope for the parallel line
Lines that are parallel have the same slope. Therefore, the slope of the line we are trying to find is also \(-\frac{3}{4}\).

### Step 3: Use the point-slope form to find the equation
The point-slope form of a line's equation is:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. We have the point \((-3, -6)\) and the slope \(m = -\frac{3}{4}\).

Substitute these values into the point-slope form:

\[
y - (-6) = -\frac{3}{4}(x - (-3))
\]

Simplifying:

\[
y + 6 = -\frac{3}{4}(x + 3)
\]

### Step 4: Simplify to slope-intercept form
Now expand the equation:

\[
y + 6 = -\frac{3}{4}x - \frac{9}{4}
\]

Subtract 6 from both sides to isolate \(y\):

\[
y = -\frac{3}{4}x - \frac{9}{4} - 6
\]

To combine the constants, convert \(6\) to a fraction with a denominator of 4:

\[
6 = \frac{24}{4}
\]

Now subtract:

\[
y = -\frac{3}{4}x - \frac{9}{4} - \frac{24}{4}
\]

\[
y = -\frac{3}{4}x - \frac{33}{4}
\]

### Final Answer:
The equation of the line in slope-intercept form is:

\[
y = -\frac{3}{4}x - \frac{33}{4}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this:

1. How do you find the slope of a line from a standard form equation?
2. What does it mean for two lines to be parallel?
3. Can you find the equation of a line perpendicular to \(3x + 4y = 7\) through the point \((-3, -6)\)?
4. How would you convert the slope-intercept form to standard form?
5. What is the significance of the \(y\)-intercept in the slope-intercept equation?

**Tip:** When dealing with parallel or perpendicular lines, always focus on how their slopes are related: parallel lines have equal slopes, while perpendicular lines have negative reciprocal slopes.

Find the equation of the line that contains the point (-3, -6) and is parallel to the line 3x + 4y = 7. Write the equation in slope-intercept form, if possible.

Discover how to find the equation of a line parallel to 3x + 4y = 7 that passes through the point (-3, -6). This detailed step-by-step solution uses algebraic methods, covering the slope-intercept form and point-slope form. Ideal for students in Grades 8-10.

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