https://file.aiquickdraw.com/mathbot/file/17276606457913xth2ibb.jpg

https://file.aiquickdraw.com/mathbot/file/1727660645791b7by3onv.jpg

Let's work through each of the steps provided in the problem:

### Step 1 of 2: Express the given equation in slope-intercept form.
The given equation is:

\[
7y - 8 = -3(4 - x)
\]

First, simplify the right-hand side by distributing the \(-3\) over \((4 - x)\):

\[
7y - 8 = -3(4) + 3x
\]
\[
7y - 8 = -12 + 3x
\]

Next, add 8 to both sides to isolate the term with \(y\):

\[
7y = 3x - 12 + 8
\]
\[
7y = 3x - 4
\]

Now, divide both sides of the equation by 7 to solve for \(y\):

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

This is the slope-intercept form of the equation, where the slope \(m = \frac{3}{7}\) and the y-intercept is \(\frac{-4}{7}\).

### Step 2 of 2: Find the equation of a line parallel to the given line that passes through the point (6, 4).

A line parallel to the given one will have the same slope, which we found to be \(\frac{3}{7}\).

Now, using the point-slope form of the equation of a line, which is:

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

Substitute \(m = \frac{3}{7}\), and the point \((x_1, y_1) = (6, 4)\):

\[
y - 4 = \frac{3}{7}(x - 6)
\]

Now, simplify:

\[
y - 4 = \frac{3}{7}x - \frac{18}{7}
\]

Add 4 (or \(\frac{28}{7}\)) to both sides to isolate \(y\):

\[
y = \frac{3}{7}x - \frac{18}{7} + \frac{28}{7}
\]
\[
y = \frac{3}{7}x + \frac{10}{7}
\]

Thus, the equation of the line in slope-intercept form is:

\[
y = \frac{3}{7}x + \frac{10}{7}
\]

---

Do you need further details on any of these steps? Here are five related questions to explore:

1. What is the significance of slope-intercept form in analyzing lines?
2. How do we determine if two lines are parallel based on their equations?
3. How can we use point-slope form for solving real-world problems?
4. What happens if two lines have the same slope but different y-intercepts?
5. How can you derive the equation of a perpendicular line instead of a parallel one?

**Tip:** In slope-intercept form, the coefficient of \(x\) is always the slope, and the constant term is the y-intercept, which helps quickly sketch or understand a line's behavior.

Consider the following equation.
7y - 8 = -3(4 - x)
Step 1 of 2: Express the given equation in slope-intercept form. Simplify your answer.
Step 2 of 2: Find the equation of the line which passes through the point (6, 4) and is parallel to the given line. Express your answer in slope-intercept form. Simplify your answer.

Learn how to express a linear equation in slope-intercept form and find the equation of a parallel line passing through a given point. This detailed solution involves key algebraic concepts such as slope-intercept form, parallel lines, and point-slope form. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation