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 determine the linear equation that represents the relationship between x and y as defined by the table provided. We need to find the slope and y-intercept from the given data points to construct the equation in the form y = mx + b.
Answer
$y = 12x - 48$
Answer for screen readers
$y = 12x - 48$
Steps to Solve
- Calculate the slope (m)
To calculate the slope $m$, we can use any two points from the table. Let's use the points (5, 12) and (6, 24). The slope is given by the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Plugging in the values: $m = \frac{24 - 12}{6 - 5} = \frac{12}{1} = 12$
- Find the y-intercept (b)
Now that we have the slope $m = 12$, we can use the slope-intercept form of a linear equation, $y = mx + b$, and one of the points from the table to solve for $b$. Let's use the point (4, 0):
$0 = 12(4) + b$ $0 = 48 + b$ $b = -48$
- Write the linear equation
Now that we have the slope $m = 12$ and the y-intercept $b = -48$, we can write the linear equation as:
$y = 12x - 48$
$y = 12x - 48$
More Information
The linear equation $y = 12x - 48$ represents the relationship between $x$ and $y$ in the given table. We found this by first calculating the slope using two points from the table, and then solving for the y-intercept.
Tips
A common mistake is incorrectly calculating the slope. Ensure you subtract the y-values and x-values in the correct order. Another mistake is incorrectly solving for the y-intercept after finding the slope. Be careful with the algebraic manipulation to isolate $b$.
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