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 find the linear equation that represents the relationship between $x$ and $y$ values given in the table. The equation should be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This question requires solving for the slope and y-intercept from the given data points.
Answer
$y = 2x + 11$
Answer for screen readers
$y = 2x + 11$
Steps to Solve
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Calculate the slope ($m$) The slope can be found using any two points from the table. Let's use (38, 87) and (40, 91). The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substitute the coordinates: $m = \frac{91 - 87}{40 - 38} = \frac{4}{2} = 2$
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Find the y-intercept ($b$) 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 (38, 87) and the slope $m = 2$. Substitute the values into the equation: $87 = 2(38) + b$ Simplify: $87 = 76 + b$ Solve for $b$: $b = 87 - 76 = 11$
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Write the linear equation Now that we have the slope $m = 2$ and the y-intercept $b = 11$, we can write the linear equation in the form $y = mx + b$. $y = 2x + 11$
$y = 2x + 11$
More Information
The equation $y = 2x + 11$ represents the linear relationship between $x$ and $y$ in the given table. For every increase of 1 in $x$, $y$ increases by 2, which is reflected in the slope of 2. The y-intercept is 11, which is the value of $y$ when $x$ is 0.
Tips
A common mistake is incorrectly calculating the slope, by either using the wrong formula e.g. $\frac{x_2 - x_1}{y_2 - y_1}$, or by incorrectly substituting the values. Another common mistake is making a mistake with the arithmatic when calculating the y intercept.
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