Question

Solve the following simultaneous equations: \[2x + y = 5\,\,;\,\,\,\, 3x - y = 5\]

Answer 1

Hint: When we try to solve the equations with two variables, it needs to be ensured that two equations are given. Then we try to convert those equations into an equation of one variable, either by elimination or substitution. Then we can get the value of one variable. We substitute the value of that variable in either of the two equations to get the value of the other variable.

Complete step-by-step solution:
Here, we are given two equations
\[ 2x + y = 5\,\,.......\,\,\left( 1 \right) \\
  3x - y = 5\,\,.......\,\,\left( 2 \right) \]
Each equation contains two variables. So, we can solve the given simultaneous equations and get the values of variables. Now, firstly we try to obtain a single equation containing one variable from given two equations. Since, variable \[y\] contain the same coefficient of opposite sign, we try to eliminate \[y\] by adding both equations. We add L.H.S. with L.H.S. and R.H.S. with R.H.S.
\[
  \,\left( {2x + y} \right) + \left( {3x - y} \right) = 5 + 5 \\
   \Rightarrow \,5x = 10 \\
   \Rightarrow x = \dfrac{{10}}{5} \\
   \Rightarrow \,x = 2 \]
Now, since we have got the value of variable \[x\] we now can get the value of variable \[y\] by putting the value of \[x\] in either of the equations.
Here, we will substitute the value of \[x\] in equation \[\left( 2 \right)\].
\[
   \Rightarrow \,3\left( 2 \right) - y = 5 \\
   \Rightarrow \,6 - y = 5 \\
   \Rightarrow \,y = 6 - 5 \\
   \Rightarrow \,y = 1 \]
Hence, we get the value of the variable \[x\] as \[2\] and value of the variable \[y\] as \[1\].

Note: We can also solve the given equations by the method of substitution. We can take all other terms except \[y\] on the right hand side in equation \[\left( 1 \right)\]. Then we get some value of\[y\], which we will substitute in equation \[\left( 2 \right)\] . Thus, we will now get equation \[\left( 2 \right)\] in form of a single variable linear equation. From \[\left( 2 \right)\] we obtain the value of \[x\] which we substitute in either equation to get the value of\[y\].