Question

What is an equation of the line in the form \[ax + by + c = 0\] with gradient \[- 2\] through the point \[(4,\ - 6)\] ?

Answer 1

Hint: In this question, we need to find the equation of the line which is in the form of \[ax+by+c=0\] . Also the gradient of the line is \[-2\] . The gradient is the same as the slope. Moreover gradients can have a positive as well as negative value. Graphically , The gradient of a horizontal line is zero and also the gradient of the \[x\] axis that is a horizontal line is zero. And also the line passes through the line \[(4,-6)\] . Here we need to find the equation of the line.
Formula used :
Gradient or Slope ,
\[m\ = \dfrac{\left({change\ in\ y} \right)}{{change\ in\ x}}\]
\[m = \dfrac{\left(y{_2}- y{_1}\right)}{\left(x{_2} – x{_1}\right)}\]

Complete step-by-step solution:
Given, gradient is \[- 2\] and the point \[(x{_1}, x{_2})\] is \[(4,\ - 6)\]
\[\Rightarrow \ m = - 2\]
We know the formula of gradient, \[m = \dfrac{\left(y{_2}- y{_1}\right)}{\left(x{_2} – x{_1}\right)}\]
Thus, \[- 2 = \dfrac{\left(y{_2}- y{_1}\right)}{\left(x{_2} – x{_1}\right)}\]
\[\Rightarrow \ - 2 = \dfrac{\left( y - \left( - 6 \right) \right)}{x – 4}\]
By cross multiplying,
We get,
\[- 2(x – 4)\ = (y – ( - 6))\]
On simplifying,
We get,
\[\Rightarrow -2x+8 = y+6\]
By moving all the terms to one side,
We get,
\[\Rightarrow - 2x-y+8–6 = 0\]
On simplifying,
We get,
\[\Rightarrow -2x – y + 2 = 0\]
\[\Rightarrow 2x+y-2 = 0\]
Thus the equation of line is \[2x+y-2 = 0\ \] which is in the form of \[ax + by + c = 0\]
Final answer :
The equation of line is \[2x + y - 2 = 0\ \] which is in the form of \[ax + by + c = 0\]

Note: The slope of a line is defined as the measure of its Steepness. It is calculated by dividing the change in \[y\] coordinate by change in \[x\] co-ordinate. Mathematically, slope is denoted by the letter \[m\]. Slope is positive when m is greater than \[0\] and when m is less than \[0\] , slope is negative. If the slope is equal to \[0\] That means it is a constant function. Graphically, The gradient of two parallel lines is equal and also the product of the gradients of two perpendicular lines is \[- 1\].