Question

which of the following is equivalent to the line defined by (4,9) and (12, -9)?
a 9x - 4y = 9
b $y - 36 = -\frac{4}{9}(x - 8)$
c $y = -\frac{9}{4}(x - 18)$
d 9x + 4y = 72

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((4,9)\) and \((12, - 9)\), we have \( x_1 = 4,y_1 = 9,x_2=12,y_2=-9 \). So \( m=\frac{-9 - 9}{12 - 4}=\frac{-18}{8}=-\frac{9}{4} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Let's use the point \((4,9)\). Substituting \( m =-\frac{9}{4}\), \( x_1 = 4\) and \( y_1 = 9\) into the point - slope form, we get \( y-9=-\frac{9}{4}(x - 4) \).

Expand this equation:
\[

$$\begin{align*} y-9&=-\frac{9}{4}x+9\\ y&=-\frac{9}{4}x + 9+9\\ y&=-\frac{9}{4}x+18\\ 4y&=- 9x + 72\\ 9x + 4y&=72 \end{align*}$$

\]

We can also check other options:

• Option A: For \( 9x-4y = 9\), when \( x = 4\), \( 9\times4-4y=9\Rightarrow36 - 4y=9\Rightarrow4y = 27\Rightarrow y=\frac{27}{4}

eq9 \), so this line does not pass through \((4,9)\).

• Option B: The slope of \( y - 36=-\frac{4}{9}(x - 8) \) is \( -\frac{4}{9}

eq-\frac{9}{4} \), so it is not the same line.

• Option C: For \( y=-\frac{9}{4}(x - 18) \), when \( x = 4\), \( y=-\frac{9}{4}(4 - 18)=-\frac{9}{4}\times(-14)=\frac{126}{4}=\frac{63}{2}

eq9 \), so this line does not pass through \((4,9)\).

Answer:

D. \( 9x + 4y=72 \)