Write the linear equation that gives the rule for the following table, write your answer as an equation with y first: x y 38 11 46 19 75 48 88 61
Understand the Problem
The question asks to find the linear equation that represents the relationship between x and y given a table of values. This involves determining the slope and y-intercept of the line that fits these points, then expressing the equation in the form y = mx + b.
Answer
$y = x - 27$
Answer for screen readers
$y = x - 27$
Steps to Solve
- Calculate the slope ($m$) using two points
We'll use the first two points $(38, 11)$ and $(46, 19)$ to find the slope:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19 - 11}{46 - 38} = \frac{8}{8} = 1$
- Use the slope and one point to find the y-intercept ($b$)
We'll use the point $(38, 11)$ and the slope $m = 1$ in the equation $y = mx + b$:
$11 = 1 \cdot 38 + b$
$11 = 38 + b$
$b = 11 - 38 = -27$
- Write the linear equation
Now that we have the slope $m = 1$ and the y-intercept $b = -27$, we can write the equation in the form $y = mx + b$:
$y = 1x - 27$
$y = x - 27$
$y = x - 27$
More Information
The equation $y = x - 27$ represents the linear relationship between $x$ and $y$ for the given data set. Each $y$ value is obtained by subtracting 27 from its corresponding $x$ value.
Tips
A common mistake is incorrectly calculating the slope or y-intercept. For example, switching the order of $y_2$ and $y_1$ or $x_2$ and $x_1$ when calculating the slope, or making an arithmetic error when solving for the y-intercept.
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