Question

If the point \[A\left( 4,1 \right)\] lies on the graph of the equation \[5x+ay=19\] , then find a.

Answer 1

Hint: In this problem, we are given the equation of a line as \[5x+ay=19\] and a point \[A\left( 4,1 \right)\] lies on it. In order to find the value of a, we first put the point in the given equation and then solve for the value of a.

Complete step by step answer:
Coordinate systems are a way to describe the position of an arbitrary point. There are various ways to describe the position of an arbitrary point and depending on the ways, we have a Cartesian, cylindrical and spherical coordinate system. In the Cartesian coordinate system, the position of a point is represented by $\left( x,y \right)$ where x is the distance from the y axis and y is the distance from the x axis. The equation of a line in the Cartesian coordinate system is a linear equation of x and y, which is of the form $ax+by=c$ .
It is given that the point \[A\left( 4,1 \right)\] lies on the given line \[5x+ay=19\] . This means that the point should satisfy the equation of the line which is \[5x+ay=19\] . This means that if we put the value of $x=4$ and $y=1$ in the equation \[5x+ay=19\] , then the LHS and RHS should match. Doing so, we get,
\[\begin{align}
  & \Rightarrow 5\left( 4 \right)+a\left( 1 \right)=19 \\
 & \Rightarrow 20+a=19 \\
 & \Rightarrow a=-1 \\
\end{align}\]
Thus, we can conclude that the value of a is $-1$ .

Note: These types of problems are pretty easy. But, if we are given some other form of the equation in some other coordinate system, then it becomes a little tricky. For example, if we are given a parametric equation instead of x and y, then, we need to put the point in the equation accordingly.