https://file.aiquickdraw.com/mathbot/file/1724420741952n0bh3x62.jpg

Let's solve the problem step by step.

### Part (a): Write the equation of the line

We are given:
- A point on the line: \((2, 5)\)
- The slope of the line: \(m = 3\)

To find the equation of a line given a point \((x_1, y_1)\) and a slope \(m\), we use the point-slope form of the line equation:

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

Substituting the given point \((x_1, y_1) = (2, 5)\) and slope \(m = 3\):

\[
y - 5 = 3(x - 2)
\]

Now, distribute the slope and simplify the equation:

\[
y - 5 = 3x - 6
\]

Add 5 to both sides to solve for \(y\):

\[
y = 3x - 1
\]

Thus, the equation of the line is:

\[
y = 3x - 1
\]

### Part (b): Sketch the graph

To sketch the graph of the line \(y = 3x - 1\), follow these steps:

1. **Identify the slope and y-intercept:**
   - Slope (\(m\)) is 3, which means the line rises 3 units for every 1 unit it moves to the right.
   - Y-intercept (\(b\)) is -1, so the line crosses the y-axis at \((0, -1)\).

2. **Plot the y-intercept:**
   - Start by plotting the point \((0, -1)\) on the graph.

3. **Use the slope to plot another point:**
   - From \((0, -1)\), move up 3 units (since the slope is 3) and 1 unit to the right. This gives you another point on the line: \((1, 2)\).

4. **Draw the line:**
   - Draw a straight line through the points \((0, -1)\) and \((1, 2)\). This line represents the equation \(y = 3x - 1\).

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

### Additional Questions for Practice:

1. What is the equation of a line passing through the point \((4, -3)\) with a slope of \(-2\)?
2. How do you convert a point-slope form equation to a slope-intercept form equation?
3. If a line has a slope of 0, what does its graph look like?
4. Can two different lines have the same slope but different y-intercepts? Explain why or why not.
5. How would you find the x-intercept of the line \(y = 3x - 1\)?

### Tip:

To find the equation of a line quickly, remember that the slope-intercept form is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

Learn how to write the equation of a line given a point and slope with a detailed step-by-step guide. This problem covers linear equations, point-slope form, and graphing lines. 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