Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
x + y = 9, 2x + y = -1

Verified step by step guidance

1. Rewrite each line in slope-intercept form \(y = mx + b\) to identify their slopes. For the first line \(x + y = 9\), solve for \(y\) to get \(y = -x + 9\). For the second line \(2x + y = -1\), solve for \(y\) to get \(y = -2x - 1\).
2. Identify the slopes of the two lines from their equations. The slope of the first line is \(m_1 = -1\), and the slope of the second line is \(m_2 = -2\).
3. Use the formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\): \(\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\) This formula gives the absolute value of the tangent of the angle between the lines.
4. Substitute the slopes \(m_1 = -1\) and \(m_2 = -2\) into the formula to find \(\tan(\theta)\): \(\tan(\theta) = \left| \frac{-1 - (-2)}{1 + (-1)(-2)} \right| = \left| \frac{-1 + 2}{1 + 2} \right|\).
5. Use a calculator to find the angle \(\theta\) by taking the arctangent (inverse tangent) of the value found in the previous step. Then, round the result to the nearest tenth of a degree to get the acute angle between the two lines.

Key Concepts

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x (rise over run). For lines given in standard form, the slope can be found by rearranging the equation into slope-intercept form (y = mx + b). Understanding slope is essential to determine the angle between two lines.

Angle Between Two Lines

The angle between two lines can be found using the formula involving their slopes: tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes. This formula gives the tangent of the acute angle between the lines, which can then be converted to degrees using an inverse tangent function.

Using a Calculator for Inverse Tangent and Rounding

After computing the tangent of the angle, use a calculator to find the inverse tangent (arctan) to get the angle in degrees. It is important to ensure the angle is acute and to round the result to the nearest tenth of a degree as specified, for precision and clarity in the final answer.