Question

1. the points given in the table lie on a line. find the slope of the line. (example 3)

x	-1	2	5	8
y	3	-1	-5	-9

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Choose two points

Let's take the first two points \((-1, 3)\) and \((2, -1)\). Here, \( x_1=-1 \), \( y_1 = 3 \), \( x_2=2 \), \( y_2=-1 \).

Step3: Calculate the slope

Substitute into the slope formula: \( m=\frac{-1 - 3}{2 - (-1)}=\frac{-4}{3} \)? Wait, no, let's check another pair. Take \((2, -1)\) and \((5, -5)\). \( x_1 = 2 \), \( y_1=-1 \), \( x_2 = 5 \), \( y_2=-5 \). Then \( m=\frac{-5 - (-1)}{5 - 2}=\frac{-4}{3} \)? Wait, no, wait the first pair: \((-1, 3)\) and \((2, -1)\). \( y_2 - y_1=-1 - 3=-4 \), \( x_2 - x_1=2 - (-1)=3 \). Wait, but let's check with \((5, -5)\) and \((8, -9)\). \( y_2 - y_1=-9 - (-5)=-4 \), \( x_2 - x_1=8 - 5=3 \). Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's take \((-1, 3)\) and \((2, -1)\). The change in y is \(-1 - 3=-4\), change in x is \(2 - (-1)=3\). Wait, but let's check the table again. x: -1, 2, 5, 8 (difference of 3 each time). y: 3, -1, -5, -9 (difference of -4 each time). So slope is \( \frac{\Delta y}{\Delta x}=\frac{-4}{3} \)? Wait, no, wait the slope formula is \( \frac{y_2 - y_1}{x_2 - x_1} \). Let's take (x1,y1)=(-1,3) and (x2,y2)=(2,-1). Then \( m=\frac{-1 - 3}{2 - (-1)}=\frac{-4}{3} \). Wait, but let's check another pair: (2,-1) and (5,-5). \( m=\frac{-5 - (-1)}{5 - 2}=\frac{-4}{3} \). And (5,-5) and (8,-9): \( m=\frac{-9 - (-5)}{8 - 5}=\frac{-4}{3} \). So the slope is \( -\frac{4}{3} \)? Wait, but that seems off. Wait, maybe I messed up the sign. Wait, 3 to -1: that's a decrease of 4, x increases by 3. So slope is -4/3. Wait, but let's confirm. The slope formula is rise over run, which is (y2 - y1)/(x2 - x1). So yes, for each pair, the change in y is -4, change in x is 3, so slope is -4/3. Wait, but let's check again. x: -1, 2 (x increases by 3), y: 3, -1 (y decreases by 4). So slope is -4/3. Yes.

Answer:

\( -\dfrac{4}{3} \)