Find the acute angle between the lines. Use degrees rounded to one decimal place.  5x-y = 8.   x+3y = 15

To find the acute angle between two lines, we use the formula for the angle \(\theta\) between two lines:

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

Where:
- \(m_1\) is the slope of the first line
- \(m_2\) is the slope of the second line

We begin by finding the slopes of the two lines:

### Step 1: Find the slope of the first line \(5x - y = 8\)
We rewrite the equation in slope-intercept form \(y = mx + b\) (where \(m\) is the slope):

\[
5x - y = 8 \implies y = 5x - 8
\]

So, the slope of the first line is \(m_1 = 5\).

### Step 2: Find the slope of the second line \(x + 3y = 15\)
Similarly, we rewrite this in slope-intercept form:

\[
x + 3y = 15 \implies 3y = -x + 15 \implies y = -\frac{1}{3}x + 5
\]

Thus, the slope of the second line is \(m_2 = -\frac{1}{3}\).

### Step 3: Use the formula for the angle between the lines
Now, we substitute \(m_1 = 5\) and \(m_2 = -\frac{1}{3}\) into the formula:

\[
\tan(\theta) = \left| \frac{5 - \left(-\frac{1}{3}\right)}{1 + 5 \cdot \left(-\frac{1}{3}\right)} \right|
\]

Simplifying the expression:

\[
\tan(\theta) = \left| \frac{5 + \frac{1}{3}}{1 - \frac{5}{3}} \right| = \left| \frac{\frac{15}{3} + \frac{1}{3}}{\frac{3}{3} - \frac{5}{3}} \right| = \left| \frac{\frac{16}{3}}{-\frac{2}{3}} \right|
\]

\[
\tan(\theta) = \left| \frac{16}{-2} \right| = 8
\]

### Step 4: Find the angle \(\theta\)
Now, we take the arctangent (inverse tangent) of 8:

\[
\theta = \tan^{-1}(8) \approx 82.9^\circ
\]

Since we need the acute angle, the acute angle is:

\[
\theta_{\text{acute}} = 90^\circ - 82.9^\circ = 7.1^\circ
\]

Thus, the acute angle between the lines is approximately **7.1 degrees**.

Would you like more details or clarifications?

---

Here are some related questions:
1. How do we calculate the angle between lines with vertical or horizontal slopes?
2. Can you find the intersection point of these two lines?
3. How does the slope affect the steepness of a line?
4. What if both lines were parallel? What would their angle be?
5. How does the formula change if the lines are perpendicular?

**Tip:** The slopes of perpendicular lines have a product of \(-1\).

This problem guides you through finding the acute angle between the lines 5x - y = 8 and x + 3y = 15 using the slope formula and trigonometric concepts. Learn step-by-step how to calculate the angle between two lines, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation