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How do you find the slope given \[ - 3x + 8y = 24\;?\]

seo-qna
Last updated date: 20th Jun 2024
Total views: 375k
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Answer
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Hint: We will put this equation into its equivalent slope intercept form of equation and then we will try to find the value of the slope.

Formula used: The general equation of slope intercept form of any equation is \[y = mx + c\], where \[m\] is the slope of the equation and \[c\] is an arbitrary constant.
Slope of any equation states the nature of the equation.
Slope of an equation is defined by the ratio of rise and run made by the straight line.
Suppose coordinates joining the line are \[({x_1},{y_1})\;and\;({x_2},{y_2})\].
So, the raise made by the line would be \[({y_2} - {y_1})\] and the run made by the line would become \[({x_2} - {x_1})\].
In such cases slope of the equation will be \[ = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}\].

Complete Step by Step Solution:
The given equation is as following:
\[ \Rightarrow - 3x + 8y = 24\].
So, we will take all the terms to the R.H.S except ‘\[8y\]’.
By doing the above operation, we get:
\[ \Rightarrow 8y = 24 + 3x\].
Now, divide both the sides of the equation by ‘\[8\]’, we get:
\[ \Rightarrow \dfrac{{8y}}{8} = \dfrac{{24 + 3x}}{8}\].
Now, by solving the above equation, we get:
\[ \Rightarrow y = \dfrac{3}{8}x + 3..............(1)\]
 Now, if we compare the equation \[(1)\] with the slope intercept form of a linear equation, we can state the following statement:
The slope intercept form of the linear equation is: \[y = mx + c\], where \[m\]is the slope.
So, by comparing it, we get:
\[ \Rightarrow m = \dfrac{3}{8}\] and \[c = 3\].

Therefore, the slope of the equation is \[\dfrac{3}{8}\].

Note: Points to remember:
The general equation of slope intercept form of any equation is \[y = mx + c\], where \[m\] is the slope of the equation and \[c\] is an arbitrary constant.
For, \[y = mx + c\]:
If we have a negative slope, the line is decreasing or falling from left to right, and passing through the point \[(0,c)\].
On the other hand, if we have a positive slope, the line is increasing or rising from left to right, and passing through the point \[(0,c)\].
From the above answer, we can infer that the line has a rise of \[3\] units vertically and a run of \[8\] units horizontally.
And, also the line will pass through the point \[(0,3)\].