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 us to find the linear equation that represents the relationship between x and y in the given table. The linear equation should be written in the form of y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
Answer
$y = 2x - 30$
Answer for screen readers
$y = 2x - 30$
Steps to Solve
- Calculate the slope (m) The slope of a linear equation can be found using two points $(x_1, y_1)$ and $(x_2, y_2)$ from the table with the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Let's use the first two points from the table, (32, 34) and (55, 80).
$m = \frac{80 - 34}{55 - 32} = \frac{46}{23} = 2$
- Calculate the y-intercept (b) Now that we have the slope $m = 2$, 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 (32, 34).
$34 = 2(32) + b$ $34 = 64 + b$ $b = 34 - 64 = -30$
- Write the linear equation We have found that the slope $m = 2$ and the y-intercept $b = -30$. We substitute these values into the slope-intercept form $y = mx + b$ to get the linear equation.
$y = 2x - 30$
$y = 2x - 30$
More Information
The linear equation $y = 2x - 30$ represents the relationship between x and y in the given table. We can verify this by plugging in the x values from the table into the equation and checking if we get the corresponding y values. For example, if $x = 62$, then $y = 2(62) - 30 = 124 - 30 = 94$, which matches the table.
Tips
A common mistake would be incorrectly calculating the slope or y-intercept. For example, swapping $x_1$ and $x_2$ or $y_1$ and $y_2$ when calculating the slope, or making an arithmetic error when solving for $b$. It is important to double-check your calculations and make sure to use consistent points when finding the slope and y-intercept.
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