Find the equation representing the relationship between x and y

To find the equation that represents the relationship between \(x\) and \(y\), we can analyze the given data points in the table.

The table provides the following pairs of \((x, y)\) values:
- \((8, 12)\)
- \((10, 14)\)
- \((11, 15)\)
- \((12.50, 16.50)\)
- \((14, 18)\)

Let's look for a linear relationship of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

First, let's find the slope \(m\) using the first two points \((8, 12)\) and \((10, 14)\):
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - 12}{10 - 8} = \frac{2}{2} = 1\)

Now that we have the slope \(m = 1\), we can use one of the points to find the y-intercept \(b\). Let's use the point \((8, 12)\):
\(12 = 1(8) + b\)
\(12 = 8 + b\)
\(b = 12 - 8 = 4\)

So the equation is \(y = 1x + 4\), which simplifies to \(y = x + 4\).

Let's check if this equation holds for the other points:
- For \((11, 15)\): \(15 = 11 + 4 = 15\) (True)
- For \((12.50, 16.50)\): \(16.50 = 12.50 + 4 = 16.50\) (True)
- For \((14, 18)\): \(18 = 14 + 4 = 18\) (True)

Since the equation \(y = x + 4\) holds true for all the given points, it represents the relationship between \(x\) and \(y\).

Final Answer: The final answer is \(\boxed{y=x+4}\)

To find the equation that represents the relationship between \(x\) and \(y\) for the given data points in the table, we can follow a similar approach as before.

The table provides the following pairs of \((x, y)\) values:
- \((-5, -8)\)
- \((-4, -7)\)
- \((-3, -6)\)
- \((-2, -5)\)
- \((-1, -4)\)

Let's look for a linear relationship of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

First, let's find the slope \(m\) using the first two points \((-5, -8)\) and \((-4, -7)\):
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-7 - (-8)}{-4 - (-5)} = \frac{-7 + 8}{-4 + 5} = \frac{1}{1} = 1\)

Now that we have the slope \(m = 1\), we can use one of the points to find the y-intercept \(b\). Let's use the point \((-5, -8)\):
\(-8 = 1(-5) + b\)
\(-8 = -5 + b\)
\(b = -8 + 5 = -3\)

So the equation is \(y = 1x - 3\), which simplifies to \(y = x - 3\).

Let's check if this equation holds for the other points:
- For \((-4, -7)\): \(-7 = -4 - 3 = -7\) (True)
- For \((-3, -6)\): \(-6 = -3 - 3 = -6\) (True)
- For \((-2, -5)\): \(-5 = -2 - 3 = -5\) (True)
- For \((-1, -4)\): \(-4 = -1 - 3 = -4\) (True)

Since the equation \(y = x - 3\) holds true for all the given points, it represents the relationship between \(x\) and \(y\).

Final Answer: The final answer is \(\boxed{y=x-3}\)