Question

How do you solve the system of equations $3x-4y=-6$ and $x-5y=9$ ?

Answer 1

Hint: We are given two equations which must be solved simultaneously as well as drawn on the same graph to find the solution of these equations. In order to plot any equation on a graph, we must compute at least two points lying on it which can be further marked and joined to sketch the graph. Therefore, we shall first find the points on these functions and then find their solutions.

Complete step by step solution:
We will put the values of x and y equal to zero one by one to find two simple points one of which will have its x-coordinate equal to zero and the other one would have its y-coordinate equal to zero.
We shall first find the points lying on the line whose equation is given by, $3x-4y=-6$.
Putting $x=0$ in the equation, we get
$3\left( 0 \right)-4y=-6$
$\Rightarrow -4y=-6$
Now, we shall take divide both sides by -4 to find y:
$\begin{align}
  & \Rightarrow y=\dfrac{-6}{-4} \\
 & \Rightarrow y=\dfrac{6}{4} \\
\end{align}$
$\therefore y=\dfrac{3}{2}$
Therefore, we get the point as $\left( 0,\dfrac{3}{2} \right)$.
Putting $y=0$ in the equation, we get
$3x-4\left( 0 \right)=-6$
$\Rightarrow 3x=-6$
Now, we shall divide both sides by 3 to find x:
$\begin{align}
  & \Rightarrow x=\dfrac{-6}{3} \\
 & \Rightarrow x=-\dfrac{2}{1} \\
\end{align}$
$\therefore x=-2$
Therefore, we get the point as (-2,0)
Hence, the points are $\left( 0,\dfrac{3}{2} \right)$and $\left( -2,0 \right)$. ……………………….. (1)
We shall now find the points on the second equation given as, $x-5y=9$.
Putting $x=0$ in the equation, we get
$0-5y=9$
Dividing both sides by -5, we get
$\Rightarrow y=-\dfrac{9}{5}$
Therefore, we get the point as $\left( 0,-\dfrac{9}{5} \right)$.
Putting $y=0$ in the equation, we get
$x-5\left( 0 \right)=9$
$\Rightarrow x=9$
Therefore, we get the point as (9,0)
Hence, the points are $\left( 0,-\dfrac{9}{5} \right)$ and $\left( 9,0 \right)$. ……………………… (2)
From (1) and (2), we get the graph as:

Therefore, the solution of the given system of equations is (-3,-6) as the two lines are intersecting at this point.

Note: While plotting the graph, the points must be marked with precision. We often tend to get confused between the two coordinates and make the points incorrectly. Also, while sketching any graph, the two points should always be taken such that the x or y coordinates are zero in them as it makes the calculations easier.