Question

the points (-23, 2) and (-7, r) lie on a line with slope \\(\frac{5}{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, \( (x_1, y_1)=(-23, 2) \), \( (x_2, y_2)=(-7, r) \), and \( m = \frac{5}{4} \).

Step2: Substitute values into formula

Substitute into the slope formula: \( \frac{5}{4}=\frac{r - 2}{-7 - (-23)} \). Simplify the denominator: \( -7 - (-23)=-7 + 23 = 16 \). So the equation becomes \( \frac{5}{4}=\frac{r - 2}{16} \).

Step3: Solve for \( r \)

Cross - multiply: \( 5\times16 = 4\times(r - 2) \). Calculate \( 5\times16 = 80 \), so \( 80 = 4r-8 \). Add 8 to both sides: \( 80 + 8=4r \), which gives \( 88 = 4r \). Divide both sides by 4: \( r=\frac{88}{4}=22 \).

Answer:

\( r = 22 \)