Find the acute angle between the lines, 2x-y=3, 3x+y=7

To find the acute angle between the two lines, we can use the formula for the angle \( \theta \) between two lines given by their slopes:

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

where \( m_1 \) and \( m_2 \) are the slopes of the two lines.

### Step 1: Find the slopes of the lines
The equation of a line in slope-intercept form is \( y = mx + c \), where \( m \) is the slope.

- For the first line \( 2x - y = 3 \), we rewrite it as:
  \[
  y = 2x - 3
  \]
  So, the slope \( m_1 = 2 \).

- For the second line \( 3x + y = 7 \), we rewrite it as:
  \[
  y = -3x + 7
  \]
  So, the slope \( m_2 = -3 \).

### Step 2: Apply the formula
Now, substitute \( m_1 = 2 \) and \( m_2 = -3 \) into the formula:

\[
\tan \theta = \left| \frac{2 - (-3)}{1 + (2)(-3)} \right| = \left| \frac{2 + 3}{1 - 6} \right| = \left| \frac{5}{-5} \right| = 1
\]

### Step 3: Find the angle
We know that \( \tan \theta = 1 \) corresponds to \( \theta = 45^\circ \).

Thus, the acute angle between the lines is \( 45^\circ \).

Would you like more details on any part of the solution?

### Related Questions:
1. How do you find the slopes of more complex lines?
2. What is the geometric interpretation of the angle between lines?
3. How do we handle lines that are parallel or perpendicular?
4. Can you explain the difference between acute and obtuse angles between lines?
5. What is the significance of the absolute value in the angle formula?

**Tip:** Always convert the line equations into slope-intercept form to easily identify the slope for calculations!

Find the acute angle between the lines, 2x - y = 3, 3x + y = 7

This problem explains how to find the acute angle between two lines, 2x - y = 3 and 3x + y = 7, using the slope-intercept form and the tangent formula for angles between lines. Includes step-by-step instructions for high school students in Grades 9-12.

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