
Find the angle between the pair of lines, round your answer to the nearest hundredth if necessary:
Line A: y=2x+3
Line B: 6x-2y=-14

To find the angle between two lines, we first need to express both lines in slope-intercept form, \(y = mx + b\), where \(m\) is the slope of the line. Once we have the slopes, we can use the following formula to find the angle \(\theta\) between the two lines:

\[
\tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1 \cdot m_2}\right|
\]

Where:
- \(m_1\) is the slope of Line A.
- \(m_2\) is the slope of Line B.

### Step 1: Find the slopes of the lines.
- **Line A:** \(y = 2x + 3\), so the slope is \(m_1 = 2\).
- **Line B:** Rewrite the equation \(6x - 2y = -14\) in slope-intercept form.

\[
6x - 2y = -14
\]
\[
-2y = -6x - 14
\]
\[
y = 3x + 7
\]

Thus, the slope of Line B is \(m_2 = 3\).

### Step 2: Use the formula to find the angle.
Now that we have the slopes \(m_1 = 2\) and \(m_2 = 3\), we can apply the formula:

\[
\tan(\theta) = \left|\frac{2 - 3}{1 + 2 \cdot 3}\right| = \left|\frac{-1}{1 + 6}\right| = \left|\frac{-1}{7}\right| = \frac{1}{7}
\]

Now, find the arctangent of \(\frac{1}{7}\):

\[
\theta = \tan^{-1}\left(\frac{1}{7}\right) \approx 8.13^\circ
\]

### Final Answer:
The angle between the two lines is approximately \(8.13^\circ\).

Would you like more details or have any questions?

Here are 5 related questions:
1. What happens when the lines are parallel? How is the angle calculated?
2. How do you calculate the angle when both lines are vertical?
3. How would the process change if given two lines in parametric form?
4. What if one of the lines had a negative slope, how would that affect the result?
5. How do you interpret the result if \(\theta\) is greater than \(90^\circ\)?

**Tip:** Always check whether the slopes are undefined (for vertical lines) before using the angle formula.

Find the angle between the pair of lines, round your answer to the nearest hundredth if necessary: Line A: y=2x+3 Line B: 6x-2y=-14

Learn how to calculate the angle between two lines using their slopes. We break down the steps to solve for the angle between y=2x+3 and 6x-2y=-14, using the slope formula and the tangent of the angle. Suitable for students in Grades 9-12.

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