To create a formula for the points \((1, 5)\), \((2, 7)\), \((3, 9)\), and \((4, 11)\), we can observe the relationship between the \(x\) and \(y\) values.
The given points can be summarized as follows:
- For \(x = 1\), \(y = 5\)
- For \(x = 2\), \(y = 7\)
- For \(x = 3\), \(y = 9\)
- For \(x = 4\), \(y = 11\)
Now let's analyze the change in \(y\):
- From \(x = 1\) to \(x = 2\), \(y\) changes from 5 to 7 (increased by 2).
- From \(x = 2\) to \(x = 3\), \(y\) changes from 7 to 9 (increased by 2).
- From \(x = 3\) to \(x = 4\), \(y\) changes from 9 to 11 (increased by 2).
The increase in \(y\) is consistent, suggesting a linear relationship. This indicates that \(y\) can be expressed as a linear equation of the form:
\[ y = mx + b \]
where \(m\) is the slope and \(b\) is the y-intercept. Since the increase is 2 for each increase of 1 in \(x\), we can determine that:
\[ m = 2 \]
Next, we can find \(b\) by substituting one of the points into the equation. We can use the point \((1,5)\):
\[ 5 = 2(1) + b \]
Solving for \(b\):
\[ 5 = 2 + b \ b = 5 - 2 \ b = 3 \]
Thus, the equation that models the given points is:
\[ y = 2x + 3 \]
This linear equation should give the correct \(y\) values for each corresponding \(x\):
- 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 \(x = 4\): \(y = 2(4) + 3 = 11\)
The formula for the points is:
\[ y = 2x + 3 \]
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