Fill in the missing numbers to complete the linear equation that gives the rule for this table, where y = ?x - ?
Understand the Problem
The question asks us to find the missing numbers in the linear equation of the form y = mx - b, given a table of x and y values. We need to determine the slope (m) and the y-intercept (b) based on the data provided in the table. We can compute the slope by taking the difference between consecutive y-values and dividing by the difference between the corresponding x-values and then substituting one of the points into y=mx-b to find the intercept.
Answer
$y = 18x - 75$
Answer for screen readers
$y = 18x - 75$
Steps to Solve
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Calculate the slope ($m$) Choose two points from the table, for example, $(4, -3)$ and $(5, 15)$. Calculate the slope using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
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Substitute the values Using the points $(4, -3)$ and $(5, 15)$, we have: $m = \frac{15 - (-3)}{5 - 4} = \frac{18}{1} = 18$
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Determine the y-intercept ($b$) Use the slope-intercept form of a linear equation: $y = mx - b$. Substitute the slope $m = 18$ and one of the points, for example, $(4, -3)$, into the equation: $-3 = 18(4) - b$
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Solve for $b$ $-3 = 72 - b$ $b = 72 + 3$ $b = 75$
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Write the complete equation Substitute the values of $m$ and $b$ into the equation $y = mx - b$. $y = 18x - 75$
$y = 18x - 75$
More Information
The linear equation represents the relationship between $x$ and $y$ values in the table. For every increase of 1 in $x$, $y$ increases by 18.
Tips
A common mistake is incorrectly calculating the slope by subtracting the $x$ and $y$ values in the wrong order. Another mistake is making an error when solving for the y-intercept $b$ after substituting the slope and a point into the equation.
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