https://file.aiquickdraw.com/mathbot/file/1727355695587app3xf46.jpg

To solve this problem, we need to find the equation of a line parallel to the given line \( y = 3x + 2 \) and passing through the point \( (10, 1) \).

### Step 1: Determine the Slope of the Given Line
The equation \( y = 3x + 2 \) is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m = 3 \).

### Step 2: Slope of the Parallel Line
A line parallel to the given line will have the same slope, so the slope of the required line is also \( 3 \).

### Step 3: Use the Point-Slope Form
We use the point-slope form of a line equation, which is:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through. Plugging in \( m = 3 \) and the point \( (10, 1) \):
\[
y - 1 = 3(x - 10)
\]
\[
y - 1 = 3x - 30
\]
\[
y = 3x - 29
\]

### Step 4: Rewrite in Standard Form
We need to rewrite this in the form \( Ax + By = C \). Subtract \( 3x \) from both sides:
\[
-3x + y = -29
\]
Multiplying by \(-1\) to match the format:
\[
3x - y = 29
\]

### Step 5: Match with the Given Options
The equation \( 3x - y = 29 \) corresponds to option **D**.

**Answer: D. \( 3x - y = 29 \)**

Would you like any further details or have any questions?

Here are five related questions you might find interesting:
1. What is the equation of a line parallel to \( y = -2x + 5 \) passing through the point \( (3, -1) \)?
2. How do you find the equation of a line perpendicular to \( y = \frac{1}{2}x - 4 \) through the point \( (-2, 3) \)?
3. What is the standard form of a line parallel to \( 4x - 3y = 12 \) passing through \( (6, 0) \)?
4. Can you derive the formula for the distance between two parallel lines?
5. How can you verify if two lines are parallel using their slopes?

**Tip:** Always ensure parallel lines have equal slopes when writing their equations!

Which of the following is the equation of a line parallel to the line y = 3x + 2, passing through the point (10,1)?

Learn how to determine the equation of a line that is parallel to a given line and passes through a specific point, utilizing the concepts of linear equations and slope. This guide includes using point-slope and standard form transformations to solve problems efficiently.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation