Questions: Write the point-slope form of the line's equation satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope = -2, passing through (-2, -3/2) What is the point-slope form of the equation of the line? (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Solution Steps

The point-slope form of a line's equation is given by:

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

where \( m \) is the slope, and \((x_1, y_1)\) is a point on the line.

We are given the slope \( m = -2 \) and the point \((-2, -\frac{3}{2})\). Substitute these values into the point-slope form:

\[ y - \left(-\frac{3}{2}\right) = -2(x - (-2)) \]

Simplify the equation:

\[ y + \frac{3}{2} = -2(x + 2) \]

The slope-intercept form of a line's equation is \( y = mx + b \). We will convert the point-slope form to this format by solving for \( y \):

\[ y + \frac{3}{2} = -2(x + 2) \]

Distribute the \(-2\):

\[ y + \frac{3}{2} = -2x - 4 \]

Subtract \(\frac{3}{2}\) from both sides to solve for \( y \):

\[ y = -2x - 4 - \frac{3}{2} \]

Convert \(-4\) to a fraction with a denominator of 2:

\[ y = -2x - \frac{8}{2} - \frac{3}{2} \]

Combine the fractions:

\[ y = -2x - \frac{11}{2} \]

Final Answer