Write the linear equation that represents the rule for the table, with y first, followed by an equals sign.
Understand the Problem
The question requires us to find the linear equation that represents the relationship between x and y as shown in the table. We need to determine the slope and y-intercept from the given data points and then write the equation in the form of y = mx + b. The final answer needs to have 'y' on the left side of the equals.
Answer
$y = 16x + 6$
Answer for screen readers
$y = 16x + 6$
Steps to Solve
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Calculate the slope ($m$) The slope of a linear equation can be found using any two points from the table. Let's use the points $(2, 38)$ and $(3, 54)$. The formula for the slope is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Substitute the given values: $$m = \frac{54 - 38}{3 - 2} = \frac{16}{1} = 16$$
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Use the point-slope form to find the equation The point-slope form is given by: $$y - y_1 = m(x - x_1)$$ We can use the slope ($m = 16$) and one of the points, for example $(2, 38)$. $$y - 38 = 16(x - 2)$$
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Simplify to slope-intercept form ($y = mx + b$) Expand the equation: $$y - 38 = 16x - 32$$ Add 38 to both sides: $$y = 16x - 32 + 38$$ $$y = 16x + 6$$
$y = 16x + 6$
More Information
The equation $y = 16x + 6$ represents the relationship between $x$ and $y$ in the given table. The slope is 16, and the y-intercept is 6.
Tips
A common mistake is incorrectly calculating the slope by swapping the x and y values in the slope formula. Another mistake is making an error when simplifying the equation from point-slope form to slope-intercept form. Double-checking calculations can help avoid these errors.
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