Question

write a rule and an equation to fit the pattern in the table.

x	y
1	24
3	26
4	27
7	30
10	33

describe the relationship in words.
the value of y is the value of x.
(type an integer or a fraction.)

Explanation:

Step1: Calculate the differences

First, we find the difference between consecutive \( y \) and \( x \) values. Let's take two points, say \((1, 24)\) and \((3, 26)\). The change in \( x \) is \( 3 - 1 = 2 \), and the change in \( y \) is \( 26 - 24 = 2 \). So the rate of change (slope) is \( \frac{2}{2}=1 \). Let's check another pair, \((3, 26)\) and \((4, 27)\). Change in \( x = 4 - 3 = 1 \), change in \( y = 27 - 26 = 1 \). Slope is \( \frac{1}{1}=1 \). Another pair: \((4, 27)\) and \((7, 30)\). Change in \( x = 7 - 4 = 3 \), change in \( y = 30 - 27 = 3 \). Slope is \( \frac{3}{3}=1 \). So the slope \( m = 1 \).

Step2: Find the y-intercept

Using the point-slope form \( y = mx + b \), we can use the point \((1, 24)\). Substitute \( x = 1 \), \( y = 24 \), and \( m = 1 \) into the equation: \( 24 = 1\times1 + b \). Solving for \( b \), we get \( b = 24 - 1 = 23 \).

Step3: Write the equation

So the equation is \( y = x + 23 \). This means the value of \( y \) is equal to the value of \( x \) plus 23.

Answer:

The value of \( y \) is equal to the value of \( x \) plus 23. The equation is \( y = x + 23 \).