Question

find an equation of the line passing through the pair of points (3,4) and (-5, -5). write the equation in the form ax + by = c. choose the correct answer below. a. -9x + 8y = 5 b. -9x + 8y = -5 c. -8x + 9y = -5 d. -8x + 9y = 5

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 \((3, 4)\) and \((-5, -5)\), we have \( x_1 = 3, y_1 = 4, x_2=-5, y_2 = -5 \). So, \( m=\frac{-5 - 4}{-5 - 3}=\frac{-9}{-8}=\frac{9}{8} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((3, 4)\) and \( m = \frac{9}{8} \), we get \( y - 4=\frac{9}{8}(x - 3) \).

Step3: Convert to standard form

First, multiply both sides by 8 to get rid of the fraction: \( 8(y - 4)=9(x - 3) \). Expand both sides: \( 8y-32 = 9x - 27 \). Then, rearrange the terms to get \( -9x + 8y=32 - 27 \), which simplifies to \( -9x + 8y = 5 \). Wait, this is wrong. Let's check again. Wait, when we expand \( 8(y - 4)=9(x - 3) \), we have \( 8y-32=9x - 27 \). Then, move all terms to the left - hand side: \( -9x+8y-32 + 27 = 0 \), so \( -9x + 8y-5 = 0 \), or \( -9x + 8y=5 \). Wait, but let's check with the other point \((-5,-5)\). Substitute \( x=-5,y = -5 \) into \( -9x + 8y \): \( -9\times(-5)+8\times(-5)=45-40 = 5 \). So the equation \( -9x + 8y = 5 \) holds for both points.

Wait, maybe I made a mistake in the sign when moving terms. Let's start over.

Slope \( m=\frac{-5 - 4}{-5 - 3}=\frac{-9}{-8}=\frac{9}{8} \).

Using point - slope form with \((3,4)\): \( y - 4=\frac{9}{8}(x - 3) \)

Multiply both sides by 8: \( 8y-32 = 9x-27 \)

Subtract \( 9x \) from both sides: \( -9x + 8y-32=-27 \)

Add 32 to both sides: \( -9x + 8y=5 \)

Now check the point \((-5,-5)\):

Left - hand side: \( -9\times(-5)+8\times(-5)=45 - 40 = 5 \), which is equal to the right - hand side.

Now check the options: Option A is \( -9x + 8y = 5 \), which matches our result.

Wait, but let's check the calculation again. Wait, when we calculated the slope, \( \frac{y_2 - y_1}{x_2 - x_1}=\frac{-5 - 4}{-5 - 3}=\frac{-9}{-8}=\frac{9}{8} \), that's correct. Then point - slope form: \( y - 4=\frac{9}{8}(x - 3) \). Multiply by 8: \( 8y-32 = 9x - 27 \). Then, \( 8y-9x=32 - 27 \), \( -9x + 8y = 5 \). Yes, that's correct.

Wait, but let's check option D: \( -8x + 9y = 5 \). Let's substitute \((3,4)\) into \( -8x + 9y \): \( -8\times3+9\times4=-24 + 36 = 12
eq5 \). Option B: \( -9x + 8y=-5 \), substitute \((3,4)\): \( -27 + 32 = 5
eq - 5 \). Option C: substitute \((3,4)\): \( -24+36 = 12
eq - 5 \). Option A: substitute \((3,4)\): \( -27 + 32 = 5 \), substitute \((-5,-5)\): \( 45-40 = 5 \). So the correct answer is A.

Answer:

A. \( -9x + 8y = 5 \)