Question

How do you solve the simultaneous equation \[2x+5y=16\] and \[4x+3y=11\]?

Answer 1

Hint: In the above type of question when there are two equation given we need to make either of the two variables same i.e. either x or y by multiplying the coefficient of the equation to the other, i.e. taking x as the variable which needs to be constant for which we will be multiplying 4 to the first equation and then multiplying 2 in the other equation then we will subtract one equation from other and we will get the value of other variable i.e. y and then we will substitute the value of y in one of the equation and we will get the value of x.

Complete step-by-step solution:
In the above question let us assume
\[2x+5y=16\ldots \ldots \ldots \text{ }\left( 1 \right)\]
\[4x+3y=11\ldots \ldots ..\text{ }\left( 2 \right)\]
Now we will make the coefficient of x same in both equations so that after subtracting we will have an equation in which that equation will be linear. So for this to happen we will be multiplying equation (1) by 4 and equation (2) by 2 and we will get:
\[\Rightarrow 8x+20y=64\ldots \ldots .\text{ }\left( 3 \right)\]
\[\Rightarrow 8x+6y=22\ldots \ldots ..\text{ }\left( 4 \right)\]
Now we will subtract equation 4 by equation 3 and we will get a linear equation in terms of y fro which we will be able to calculate the value of y and we will get:
\[\begin{align}
  & \Rightarrow 14y=42 \\
 & \Rightarrow y=3 \\
\end{align}\]
Now we will substitute the value of y in equation 1 and we will get the value of x
\[\begin{align}
  & \Rightarrow 2x+5\left( 3 \right)=16 \\
 & \Rightarrow 2x=16-15 \\
 & \Rightarrow 2x=1 \\
 & \Rightarrow x=\dfrac{1}{2} \\
\end{align}\]
Now we have got the values of x and y which will be satisfying both the equation as \[x=\dfrac{1}{2}\] and \[y=3\] .

Note: In the above type of question always remember to make one of the variable in the given two equation constant so as to form a linear equation with the other variable like we did in this question, we took x as constant variable and formed a linear equation in y and then substituted in one of the equation that was in the question to find the value x. We can also go for the method of substitution to solve the equations.