Fill in the missing numbers to complete the linear equation that gives the rule for this table.
Understand the Problem
The question asks to find the missing numbers in the linear equation that represents the relationship between x and y in the given table. You need to determine the slope and y-intercept of the linear equation, then fill these values in the equation y = mx + b.
Answer
$y = 9x + 0$
Answer for screen readers
$y = 9x + 0$
Steps to Solve
- Find the slope ($m$)
The slope $m$ represents the rate of change of $y$ with respect to $x$. We can calculate it using any two points from the table. Let's use the points $(3, 27)$ and $(4, 36)$.
$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{36 - 27}{4 - 3} = \frac{9}{1} = 9 $$
- Find the y-intercept ($b$)
The y-intercept $b$ is the value of $y$ when $x = 0$. 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 $(3, 27)$ and the slope $m = 9$.
$$ 27 = 9(3) + b $$ $$ 27 = 27 + b $$ $$ b = 27 - 27 = 0 $$
- Write the linear equation
Now that we have the slope $m = 9$ and the y-intercept $b = 0$, we can write the linear equation in the form $y = mx + b$.
$$ y = 9x + 0 $$ $$ y = 9x $$
$y = 9x + 0$
More Information
The equation describes a direct proportionality between $x$ and $y$. For every unit increase in $x$, $y$ increases by 9 units. The line passes through the origin (0,0).
Tips
A common mistake is incorrectly calculating the slope or swapping the $x$ and $y$ values in the slope formula. Another mistake is incorrectly solving for the y-intercept after finding the slope. A final mistake is assuming the y-intercept is always present in the table
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