Question

create a formula for the points (1, 5), (2, 7), (3, 9) and (4, 11)

Answer 1

To find a formula for the points (1, 5), (2, 7), (3, 9), and (4, 11), we can observe the pattern in the \( y \)-coordinates with respect to \( x \)-coordinates.

Let's list the points and their coordinates:

- For \( x = 1 \), \( y = 5 \)
- For \( x = 2 \), \( y = 7 \)
- For \( x = 3 \), \( y = 9 \)
- For \( x = 4 \), \( y = 11 \)

We can see that as \( x \) increases by 1, \( y \) increases by 2. This suggests a linear relationship of the form:

\[
y = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept.

1. To find the slope \( m \):

Using the change in \( y \) over the change in \( x \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{2 - 1} = \frac{2}{1} = 2
\]

So \( m = 2 \).

2. Now we need to find \( b \). We can use one of the points to solve for \( b \). Let's use the point \( (1, 5) \).

Substituting \( x = 1 \) and \( y = 5 \) into the equation:

\[
5 = 2(1) + b
\]
\[
5 = 2 + b
\]
\[
b = 5 - 2 = 3
\]

Now we have the values for \( m \) and \( b \). The linear equation that fits the given points is:

\[
y = 2x + 3
\]

You can verify this formula by substituting the values of \( x \) into the equation to see if you get the corresponding \( y \) values:

- For \( x = 1 \): \( y = 2(1) + 3 = 5 \)
- For \( x = 2 \): \( y = 2(2) + 3 = 7 \)
- For \( x = 3 \): \( y = 2(3) + 3 = 9 \)
- For \( x = 4 \): \( y = 2(4) + 3 = 11 \)

Thus, the formula for the points is:

\[
y = 2x + 3
\]