Question

How do you solve the system \[2x+4y=8\] and \[-5x+3y=5\] ?

Answer 1

Hint: These types of problems are pretty straight forward and are very easy to solve. For such types of problems, we need to have a clear cut idea about linear equations and simultaneous equations. In problems like these we need to find the value of the unknown parameters of the equation, which are ‘x’ and ‘y’. We need to multiply any of the above equations with a suitable integer such that on adding or subtracting the equations, one the unknown parameter gets eliminated and we can find the value of the other. After finding this value, we put the value of the known parameter in any of the two equations to find the value of the other.

Complete step by step solution:
Now we start off with the solution to the problem by writing the two equations as,
\[2x+4y=8\] ---- Equation \[1\]
\[-5x+3y=5\] ----- Equation \[2\]
We now multiply the first equation with \[3\] on both the sides and it transforms to,
\[6x+12y=24\] ---- Equation \[3\]
We multiply the second equation with \[4\] on both the sides and it transforms to,
\[-20x+12y=20\] ---- Equation \[4\]
We now subtract the fourth equation from the third and we get,
\[6x+12y-\left( -20x+12y \right)=24-20\]
On evaluating we get,
\[\begin{align}
  & 6x+12y-\left( -20x+12y \right)=24-20 \\
 & \Rightarrow 6x+12y+20x-12y=4 \\
 & \Rightarrow 26x=4 \\
 & \Rightarrow x=\dfrac{4}{26} \\
 & \Rightarrow x=\dfrac{2}{13} \\
\end{align}\]
Now, putting this value of \[x\] in equation \[1\] we find the value of \[y\] as,
\[\begin{align}
  & 2x+4y=8 \\
 & \Rightarrow 2\times \dfrac{2}{13}+4y=8 \\
 & \Rightarrow 4y=8-\dfrac{4}{13} \\
 & \Rightarrow 4y=\dfrac{100}{13} \\
 & \Rightarrow y=\dfrac{25}{13} \\
\end{align}\]

Thus the solution to our problem is \[x=\dfrac{2}{13}\] and \[y=\dfrac{25}{13}\].

Note: For these types of problems we need to remember the theory of linear and simultaneous equations chapters very well. The problem can also be solved using the method of graphs. In such a method we need to transform both the linear equations in the form of equations of straight lines and then plot them on the graph paper. After doing so, we need to find the point of intersection, and this point lies on both the lines simultaneously and gives the required value of ‘x’ and ‘y’.