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: Check each option

- **Option A**: Rewrite \( 9x-4y = 9 \) in slope - intercept form \( y=mx + b \). We get \( 4y=9x - 9\Rightarrow y=\frac{9}{4}x-\frac{9}{4} \). The slope is \( \frac{9}{4} eq-\frac{9}{4} \), so A is incorrect.
- **Option B**: The equation \( y - 36=-\frac{4}{9}(x - 8) \) has a slope of \( -\frac{4}{9} eq-\frac{9}{4} \), so B is incorrect.
- **Option C**: The equation \( y=-\frac{9}{4}(x - 18) \) is in point - slope form \( y - y_0=m(x - x_0) \) with slope \( m = -\frac{9}{4} \). Let's check if the points lie on this line. For \( x = 4 \), \( y=-\frac{9}{4}(4 - 18)=-\frac{9}{4}\times(-14)=\frac{126}{4}=\frac{63}{2} eq9 \). Wait, maybe we made a mistake. Let's find the equation of the line using the two points. Using point - slope form with point \((4,9)\) and slope \( -\frac{9}{4} \), the equation is \( y - 9=-\frac{9}{4}(x - 4) \), which simplifies to \( y=-\frac{9}{4}x+9 + 9=-\frac{9}{4}x + 18 \). Now, let's check option D.
- **Option D**: Rewrite \( 9x+4y = 72 \) in slope - intercept form. \( 4y=-9x + 72\Rightarrow y=-\frac{9}{4}x+18 \). Now check if the points \((4,9)\) and \((12,-9)\) lie on this line. For \( x = 4 \), \( y=-\frac{9}{4}\times4+18=-9 + 18 = 9 \). For \( x = 12 \), \( y=-\frac{9}{4}\times12+18=-27 + 18=-9 \). So the line \( 9x + 4y=72 \) passes through both points. But the original option C was marked, maybe there was a mistake in the initial check. Wait, the user's marked option C: Let's re - evaluate. The equation \( y=-\frac{9}{4}(x - 18) \) can be rewritten as \( y=-\frac{9}{4}x+\frac{162}{4}=-\frac{9}{4}x+\frac{81}{2} \). When \( x = 4 \), \( y=-\frac{9}{4}\times4+\frac{81}{2}=-9+\frac{81}{2}=\frac{-18 + 81}{2}=\frac{63}{2} eq9 \). So there must be a mistake. Wait, the correct equation of the line through \((4,9)\) and \((12,-9)\) is \( y=-\frac{9}{4}x + 18 \), which is equivalent to \( 9x+4y = 72 \) (multiplying both sides by 4: \( 4y=-9x + 72\Rightarrow9x + 4y=72 \)). But the options given: Let's check option D again. Option D is \( 9x + 4y=72 \), which we saw passes through both points. But the user's marked option was C. Maybe there was a typo. However, based on the slope and the points, the correct equivalent line is \( 9x + 4y = 72 \) (Option D). But since the user's marked option was C, maybe we misread the options. Wait, the original problem's option C: \( y=-\frac{9}{4}(x - 18) \), if we expand it: \( y=-\frac{9}{4}x+\frac{162}{4}=-\frac{9}{4}x+\frac{81}{2} \). The line we found is \( y=-\frac{9}{4}x + 18=-\frac{9}{4}x+\frac{72}{4} \). These are different. So there must be an error. But according to the calculation of the slope and the line equation, the correct equivalent line is \( 9x + 4y = 72 \) (Option D). But the user's marked option was C, maybe the question had a mistake. However, if we re - check the slope of option C, it has the correct slope \( -\frac{9}{4} \), but the point does not match. Wait, maybe the intended answer is D, but the user marked C. But according to the calculation, the correct answer is D.
Wait, we made a mistake in the initial analysis of option C. Let's substitute \( x = 18 \) into the line equation we found \( y=-\frac{9}{4}x + 18 \). When \( x = 18 \), \( y=-\frac{9}{4}\times18+18=-\frac{162}{4}+18=-\frac{81}{2}+18=-\frac{81}{2}+\frac{36}{2}=-\frac{45}{2} eq0 \). So option C is incorrect. Option D: \( 9x + 4y = 72 \), when \( x = 4 \), \( 9\times4+4y = 72\Rightarrow36 + 4y=72\Rightarrow4y = 36\Rightarrow y = 9 \). When \( x = 12 \), \( 9\times12+4y = 72\Rightarrow108+4y = 72\Rightarrow4y=-36\Rightarrow y=-9 \). So option D is correct.

Answer:

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