Fill in the missing numbers to complete the linear equation that gives the rule for this table: y = []x - []
Understand the Problem
The question provides a table of x and y values, and asks us to determine the linear equation that represents the relationship between x and y. We need to find the missing numbers in the equation y = []x - [].
Answer
$y = 1x - 3$
Answer for screen readers
$y = 1x - 3$
Steps to Solve
- Find the slope
To find the slope, we look at the change in $ y $ divided by the change in $ x $. Using the points $ (4, 1) $ and $ (5, 2) $, we get:
$$ m = \frac{2 - 1}{5 - 4} = \frac{1}{1} = 1 $$ Therefore the slope is 1.
- Write the equation in point-slope form
The point-slope form of a linear equation is: $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is a point on the line. Substituting $ m = 1 $ and the point $ (4, 1) $, we get:
$ y - 1 = 1(x - 4) $
- Convert to slope-intercept form
We want to write the equation in the form $ y = mx + b $. From the previous step, we have:
$ y - 1 = 1(x - 4) $ $ y - 1 = x - 4 $ $ y = x - 4 + 1 $ $ y = x - 3 $
Therefore, the equation is $ y = 1x - 3 $.
$y = 1x - 3$
More Information
The linear equation representing the relationship between $x$ and $y$ in the table is $y = x - 3$. This means for every increase of 1 in $x$, $y$ also increases by 1, and when $x$ is 0, $y$ is -3.
Tips
A common mistake is to incorrectly calculate the slope by reversing the order of subtraction in the numerator or denominator. Another common mistake is to incorrectly distribute or simplify when converting from point-slope form to slope-intercept form, leading to an incorrect y-intercept. Always double-check your arithmetic to avoid these mistakes.
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