Question

How do you solve the system \[2x+7y=3\] and \[x=1-4y\] ?

Answer 1

Hint: The above two mentioned equations are that of a pair of simultaneous equations which are generally used to solve for the parameters \[x\] and \[y\] . These two sets of linear equations are independent of each other and once solved gives a real value of \[x\] and \[y\] . We express one equation in terms of \[x\] or \[y\] and put this value of \[x\] or \[y\] in the other equation. Thus we find the value of one parameter and putting this value in the original equation we find the value of another.

Complete step by step answer:
The given system of linear equations is
\[2x+7y=3\] ---- Equation \[1\]
\[x=1-4y\] ----- Equation \[2\]
Now, putting the value of x from equation \[2\] and putting it in equation \[1\] , we get,
\[\begin{align}
  & 2\left( 1-4y \right)+7y=3 \\
 & \Rightarrow 2-8y+7y=3 \\
 & \Rightarrow 2-y=3 \\
 & \Rightarrow 2-3=y \\
 & \Rightarrow y=-1 \\
\end{align}\]
Now we can put this value of \[y\] either in equation \[1\] or in equation \[2\] , to get the value of \[x\] . Putting this value of \[y\] in equation \[2\] we get,
\[\begin{align}
  & x=1-4\left( -1 \right) \\
 & \Rightarrow x=1+4 \\
 & \Rightarrow x=5 \\
\end{align}\]

Thus on solving the given two equations we get \[x=5\] and \[y=-1\] .

Note: We can also solve these types of simultaneous equations or pairs of linear equations in any of the two types mentioned below. However the solution covers the simplest method of solving such equations.
Method 1: We can multiply the two equations with a suitable set of integers, and then by performing addition or subtraction we can eliminate one parameter and find the value of another. Now by putting the value of the parameter in the original equation, we can find the value of the other unknown variable.
Method 2: If we consider the graphical method of solving linear equations, then we have to plot both the equations in the 2-D coordinate system. The Intersection point of the two linear equations gives us the results of the required values of \[x\] and \[y\] .