Question

assuming a linear relationship, find the missing value in the table below. \

$$\begin{tabular}{|c|c|c|c|c|c|} \\hline $x$ & 1 & 2 & 3 & 4 & 5 \\\\ \\hline $y$ & 6 & 13 & 20 & 27 & \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Find the common difference (slope)

To find the slope \( m \) of the linear relationship, we use the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Taking two points, e.g., \( (1, 6) \) and \( (2, 13) \), we get \( m=\frac{13 - 6}{2 - 1}=\frac{7}{1} = 7 \).

Step2: Determine the linear equation

Using the point - slope form \( y - y_1=m(x - x_1) \) with the point \( (1,6) \) and \( m = 7 \), we have \( y-6=7(x - 1) \). Simplifying, \( y-6 = 7x-7 \), so \( y=7x - 1 \).

Step3: Find the missing value

When \( x = 5 \), substitute into the equation \( y=7x - 1 \). So \( y=7\times5-1=35 - 1=34 \). We can also check by looking at the pattern in the \( y \)-values: \( 6,13,20,27 \), each time we add 7. So \( 27 + 7=34 \).

Answer:

34