Question

How do you solve $3x+4y=4$ and $2x+y=6$ ?

Answer 1

Hint: We are given two linear equations in two variables which have to be solved simultaneously in order to find the solution of the given system of equations. We shall begin with finding two points lying on each line to plot their respective graphs. This will be done easily by substituting x and y equal to zero one-by-one in the equation. Then we shall mark the point of intersection of the straight-lines as their solution.

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, $3x+4y=4$.
Putting $x=0$ in the equation, we get
$3\left( 0 \right)+4y=4$
$\Rightarrow 4y=4$
Now, we shall take divide both sides by 4 to find y:
$\begin{align}
  & \Rightarrow y=\dfrac{4}{4} \\
 & \Rightarrow y=1 \\
\end{align}$
Therefore, we get the point as (0,1)
Putting $y=0$ in the equation, we get
$3x+4\left( 0 \right)=4$
$\Rightarrow 3x=4$
Now, we shall divide both sides by 3 to find x:
$\Rightarrow x=\dfrac{4}{3}$
Therefore, we get the point as $\left( \dfrac{4}{3},0 \right)$
Hence, the points are $\left( 0,1 \right)$and $\left( \dfrac{4}{3},0 \right)$. …………………….. (1)
We shall now find the points on second equation given as, $2x+y=6$
Putting $x=0$ in the equation, we get
$\begin{align}
  & 2\left( 0 \right)+y=6 \\
 & \Rightarrow y=6 \\
\end{align}$
Therefore, we get the point as (0,6).
Putting $y=0$ in the equation, we get
$\begin{align}
  & 2x+\left( 0 \right)=6 \\
 & \Rightarrow 2x=6 \\
\end{align}$
Dividing both sides by 2, we get
$\begin{align}
  & \Rightarrow x=\dfrac{6}{2} \\
 & \Rightarrow x=3 \\
\end{align}$
Therefore, we get the point as (3,0).
Hence, the points are $\left( 0,6 \right)$and $\left( 3,0 \right)$ . ………………… (2)
From (1) and (2), we get the graph as:

Therefore, the solution of the given system of equations is (4,-2) as it is the point of intersection of the two straight-lines.

Note: 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. While plotting the graph, the points must be marked with precision. One possible mistake we could have done was marking (0, -6) instead of (0,6).