Question

assuming a linear relationship, find the missing value in the table below.
x | 1 | 2 | 3 | 4 | 5
y | 7 | 16 | 25 | 34 |
(the table is presented in a tabular form with x and y rows, and columns for x values 1 - 5 and corresponding y values with the last y value missing)

Explanation:

Step1: Find the slope (rate of change)

To find the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the first two points \((1, 7)\) and \((2, 16)\). Then \( m=\frac{16 - 7}{2 - 1}=\frac{9}{1} = 9 \). We can check with other points, say \((2, 16)\) and \((3, 25)\): \( m=\frac{25 - 16}{3 - 2}=\frac{9}{1}=9 \). So the slope is 9.

Step2: Find the equation of the line

Using the point - slope form \( y - y_1=m(x - x_1) \). Using the point \((1,7)\) and \( m = 9 \), we have \( y-7 = 9(x - 1) \). Simplify this equation: \( y-7=9x - 9\), then \( y=9x - 9 + 7=9x-2 \).

Step3: Find the missing value

We need to find the value of \( y \) when \( x = 5 \). Substitute \( x = 5 \) into the equation \( y=9x-2 \). So \( y=9\times5-2=45 - 2=43 \). We can also find it by using the pattern in the \( y \) - values. Since the common difference between consecutive \( y \) - values is 9 (16 - 7 = 9, 25 - 16 = 9, 34 - 25 = 9), to find the \( y \) - value when \( x = 5 \), we add 9 to the \( y \) - value when \( x = 4 \). The \( y \) - value when \( x = 4 \) is 34, so \( 34+9 = 43 \).

Answer:

43