Write the linear equation that gives the rule for this table where the table contains the points (6, -30), (27, -9), (48, 12), and (69, 33). Write your answer as an equation with y... Write the linear equation that gives the rule for this table where the table contains the points (6, -30), (27, -9), (48, 12), and (69, 33). Write your answer as an equation with y first, followed by an equals sign.

Understand the Problem

The question asks us to find the linear equation (y = mx + b) that represents the relationship between x and y values given in the table. To do this, we need to calculate the slope (m) and the y-intercept (b) using the coordinates provided. The final answer should be written in the form 'y = ...'.

Answer

$y = x - 36$

Steps to Solve

Calculate the slope (m)

The slope $m$ of a linear equation can be found using any two points $(x_1, y_1)$ and $(x_2, y_2)$ from the table:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's use the first two points from the table, $(6, -30)$ and $(27, -9)$:

$m = \frac{-9 - (-30)}{27 - 6} = \frac{-9 + 30}{21} = \frac{21}{21} = 1$

Calculate the y-intercept (b)

The y-intercept $b$ can be found by substituting the slope $m$ and one of the points $(x, y)$ into the linear equation $y = mx + b$ and solving for $b$.

Let's use the point $(6, -30)$ and the slope $m = 1$:

$-30 = 1(6) + b$ $-30 = 6 + b$ $b = -30 - 6$ $b = -36$

Write the linear equation

Now that calculate the slope $m = 1$ and the y-intercept $b = -36$, we can write the linear equation $y = mx + b$:

$y = 1x - 36$ $y = x - 36$

$y = x - 36$

More Information

The linear equation represents the relationship between $x$ and $y$ values in the table such that for every increase of 1 in the $x$ value (21 in this case), the $y$ value increases by 1 (21 in this case). The line intercepts the y-axis at $y = -36$.

Tips

A common mistake is to incorrectly calculate the slope or to mix up the x and y values when calculating the slope. Another mistake is to make an error when solving for the y-intercept after correctly calculating the slope. To avoid these mistakes double check your math.

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