Question

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

Answer Options:

y=−2x+3
y=−2x+7
y=2x+3
y=2x−3

Answer 1

To determine the correct formula for the points (1, 5), (2, 7), (3, 9), and (4, 11), we start by observing the pattern in the coordinates.

We can first check if the relationship between x and y is linear. Let's find the slope \( m \) between a couple of points:

Using points (1, 5) and (2, 7):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{2 - 1} = \frac{2}{1} = 2
\]

Using points (2, 7) and (3, 9):

\[
m = \frac{9 - 7}{3 - 2} = \frac{2}{1} = 2
\]

Using points (3, 9) and (4, 11):

\[
m = \frac{11 - 9}{4 - 3} = \frac{2}{1} = 2
\]

In all cases, the slope \( m \) is consistently 2, indicating a linear relationship.

The general formula for a line is:

\[
y = mx + b
\]

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

We know \( m = 2 \), so we can substitute this into our formula:

\[
y = 2x + b
\]

Next, to find \( b \), we can substitute one of the points. Let's use the point (1, 5):

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

So the equation of the line is:

\[
y = 2x + 3
\]

Now, let's verify that this equation holds for the other points:

- For \( x = 2 \): \( y = 2(2) + 3 = 4 + 3 = 7 \) (matches the point (2, 7))
- For \( x = 3 \): \( y = 2(3) + 3 = 6 + 3 = 9 \) (matches the point (3, 9))
- For \( x = 4 \): \( y = 2(4) + 3 = 8 + 3 = 11 \) (matches the point (4, 11))

The equation holds true for all the given points.

Thus, the correct formula from the provided options is:

\[
\boxed{y = 2x + 3}
\]