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How do you find the linear function, \[f\left( x \right)=mx+b\], whose graph has the slope \[\dfrac{-17}{13}\] and y – intercept is (0, -4)?

Last updated date: 12th Jul 2024
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Hint: Assume \[f\left( x \right)=y\] and consider the relation \[y=mx+b\] as the linear function to be found. Consider ‘m’ as the slope and ‘b’ as the y – intercept and substitute the value of m equal to \[\left( \dfrac{-17}{13} \right)\] and ‘b’ equal to (-4) to get the required linear relation.

Complete step-by-step solution:
Here, we have been provided with the information that slope of the graph of a linear function \[f\left( x \right)=mx+b\] is \[\dfrac{-17}{13}\] and y – intercept of this graph is given as (0, -4). We are asked to find this linear relation \[f\left( x \right)=mx+b\], that means we need to find the values of ‘m’ and ‘b’ and substitute them in the given linear relation.
Now, let us assume \[f\left( x \right)=y\], i.e., y as a function of x, so we have the relation as:-
\[\Rightarrow y=mx+b\]
We know that this is the slope – intercept form of a linear equation where the coefficient of x, i.e., m, is the slope of the line and b is the y – intercept of the line. Here, y – intercept means the point where the line cut the y – axis.
Now, we have been provided with the value of slope equal to \[\dfrac{-17}{13}\], so we have,
\[\Rightarrow m=\left( \dfrac{-17}{13} \right)\]
Also, the coordinates of y – intercept is given to us as (0, -4). As we can see that the x – coordinate is 0 that means at this point the line is cutting the y – axis. Since, the y – coordinate is (-4) which means y – intercept is negative. This signifies that the line is cutting the y – axis at a distance of 4 units below the origin.
\[\Rightarrow b=\left( -4 \right)\]
Now, substituting the obtained values of m and b in the linear function, we get,
\[\Rightarrow y=\left( \dfrac{-17}{13} \right)x+\left( -4 \right)\]
\[\Rightarrow y=\dfrac{-17x}{13}-4\]
\[\Rightarrow f\left( x \right)=\dfrac{-17x}{13}-4\]
Hence, the above linear function is our answer.

Note: One may note that we can easily derive other forms of the straight line, like: - point – slope form, standard form, intercept form etc, from the above obtained slope – intercept form of the line. The main thing is that you must remember the general equations of other forms of the line. In the above question substitute the value of ‘b’ with the sign given otherwise you will get the wrong equation.