Question

the points (6, r) and (10, -6) lie on a line with slope $-\frac{1}{4}$. find the missing coordinate r.

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1,y_1)=(6,r) \), \( (x_2,y_2)=(10, - 6) \), and \( m =-\frac{1}{4} \).

Step2: Substitute values into formula

Substitute into the slope formula: \( -\frac{1}{4}=\frac{-6 - r}{10 - 6} \). Simplify the denominator: \( 10 - 6 = 4 \), so we have \( -\frac{1}{4}=\frac{-6 - r}{4} \).

Step3: Solve for \( r \)

Multiply both sides by 4: \( - 1=-6 - r \). Add 6 to both sides: \( -1 + 6=-r \), which simplifies to \( 5=-r \). Multiply both sides by - 1: \( r=-5 \).

Answer:

\( r = - 5 \)