Question

name
*you must show your work
write a linear function that relates y to x.

1. (10, -1) and (20, 9)

2.

x	y
-4	10
1	70
6	30
11	-10

Explanation:

Step1: Find the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \((10, - 1)\) and \((20,9)\), we have \(x_1 = 10,y_1=-1,x_2 = 20,y_2 = 9\). So \(m=\frac{9-(-1)}{20 - 10}=\frac{10}{10}=1\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Using the point \((10,-1)\) and \(m = 1\), we get \(y-(-1)=1\times(x - 10)\), which simplifies to \(y + 1=x - 10\), and then \(y=x-11\).

For the table (second sub - question):

Step1: Find the slope (m)

Take two points from the table, say \((-4,10)\) and \((1,70)\). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we have \(x_1=-4,y_1 = 10,x_2=1,y_2 = 70\). So \(m=\frac{70 - 10}{1-(-4)}=\frac{60}{5}=12\). Wait, there seems to be a mistake. Let's take another pair, say \((1,70)\) and \((6,30)\). Then \(m=\frac{30 - 70}{6 - 1}=\frac{- 40}{5}=-8\). Wait, maybe I misread the table. Let's check the table again. If the table is:

x	y
-4	10
1	70
6	30
11	- 10

Wait, let's recalculate the slope between \((-4,10)\) and \((6,30)\): \(m=\frac{30 - 10}{6-(-4)}=\frac{20}{10} = 2\). No, that's not right. Wait, maybe the table is:
Wait, maybe the correct pairs: Let's take \((-4,10)\) and \((1,70)\): \(m=\frac{70 - 10}{1-(-4)}=\frac{60}{5}=12\). Then \((1,70)\) and \((6,30)\): \(m=\frac{30 - 70}{6 - 1}=\frac{-40}{5}=-8\). This is inconsistent. Wait, maybe the table is written incorrectly or I misread it. Assuming the first sub - question (points \((10,-1)\) and \((20,9)\)):

Answer:

For the points \((10,-1)\) and \((20,9)\), the linear function is \(y = x-11\).

(Note: There seems to be an error in the table values as the slope is not consistent. If we assume the table is correct, we need to re - evaluate. But based on the first sub - question, the answer is \(y=x - 11\))