Question

1. what is the value of \\(k\\) such that the line passing through \\((4, -5)\\) and \\((2, k)\\) is parallel to the line \\(y = -4x + 3\\)?

Explanation:

Identify the given line's slope

Using the Coordinate Geometry knowledge point, we find the slope of the reference line.
The given line equation is:

$$y = -4x + 3$$

This is in slope-intercept form \(y = mx + b\), where the slope \(m_1\) is:

$$m_1 = -4$$

Determine the parallel slope condition

Parallel lines must have equal slopes.
Therefore, the slope \(m_2\) of the line passing through \((4, -5)\) and \((2, k)\) must be:

$$m_2 = -4$$

Set up the slope formula

Using the Coordinate Geometry slope formula for two points \((x_1, y_1)\) and \((x_2, y_2)\):

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the points \((4, -5)\) and \((2, k)\):

$$-4 = \frac{k - (-5)}{2 - 4}$$

Solve for the unknown value k

Simplify the denominator and solve the equation:

$$-4 = \frac{k + 5}{-2}$$

Multiply both sides by \(-2\):

$$8 = k + 5$$

Subtract \(5\) from both sides:

$$k = 3$$

Answer:

\(k = 3\)