Write the linear equation that gives the rule for this table. Write your answer as an equation with y first, followed by an equals sign.
Understand the Problem
The question asks to derive a linear equation from a table of x and y coordinate pairs, it requires calculating the slope and intercept.
Answer
$y = x - 35$
Answer for screen readers
$y = x - 35$
Steps to Solve
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Calculate the slope using two points
We can pick any two points from the table to calculate the slope. Let's use $(2, -33)$ and $(9, -26)$. The slope $m$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Substituting the coordinates: $$m = \frac{-26 - (-33)}{9 - 2} = \frac{-26 + 33}{7} = \frac{7}{7} = 1$$
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Find the y-intercept
Now that we have the slope $m = 1$, we can use the point-slope form of a linear equation: $$y - y_1 = m(x - x_1)$$ Using the point $(2, -33)$ and $m = 1$: $$y - (-33) = 1(x - 2)$$ $$y + 33 = x - 2$$ $$y = x - 2 - 33$$ $$y = x - 35$$ So, the y-intercept is $-35$.
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Write the linear equation The linear equation is in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We found that $m = 1$ and $b = -35$. Therefore, the equation is: $$y = 1x - 35$$ Which simplifies to: $$y = x - 35$$
$y = x - 35$
More Information
The equation $y = x - 35$ represents the relationship between $x$ and $y$ values in the given table. For every increase of 1 in $x$, $y$ also increases by 1, and when $x$ is 0, $y$ is -35.
Tips
- Incorrect slope calculation: A common mistake is to mix up the coordinates when calculating the slope, e.g., calculating $(x_2 - x_1) / (y_2 - y_1)$ instead of $(y_2 - y_1) / (x_2 - x_1)$.
- Sign errors: Be careful with negative signs, especially when subtracting negative numbers.
- Algebraic errors: Make sure to correctly isolate $y$ when solving for the y-intercept.
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