Question

How do you solve using linear combination: $3x - y = 4$, $2x + y = 6$?

Answer 1

Hint: First, multiply the equations so as to make the coefficients of the variables to be eliminated equal. Next, add or subtract the equation the equations obtained according as the terms having the same coefficients are of opposite or of the same sign. Solve the equation in one variable. Substitute the value found in any of the equations after substitution and find the value of another variable. The values of the variables constitute the solution of the given system of equations.

Complete step-by-step solution:
In this method, we eliminate one of the two variables to obtain an equation in one variable which can be easily solved. Putting the value of this variable in any one of the given equations, the value of another variable can be obtained.
Step by step solution:
The given system of equations is
$3x - y = 4$…(i)
$2x + y = 6$…(ii)
Now, multiply the equations so as to make the coefficients of the variables to be eliminated equal.
Here, the coefficients of the variable $y$ are already equal.
Next, add or subtract the equation the equations obtained according as the terms having the same coefficients are of opposite or of the same sign.
We can eliminate variable $y$ by adding equations (i) and (ii).
$5x = 10$
Divide both sides of equation by $5$, we get
$\therefore x = 2$
Now, substitute the value of $x$ in equation (ii) and find the value of $y$.
$4 + y = 6$
Subtract $4$ to both sides of the equation, we get
$\therefore y = 2$
Hence, the solution of the given system of equations is $x = 2$, $y = 2$.

Note: We can also find the solution of a given system by Method of Cross-Multiplication.
System of equations:
$3x - y = 4$…(i)
$2x + y = 6$…(ii)
By cross-multiplication, we have
$\dfrac{x}{{\begin{array}{*{20}{c}}
  { - 1}&{ - 4} \\
  1&{ - 6}
\end{array}}} = \dfrac{{ - y}}{{\begin{array}{*{20}{c}}
  3&{ - 4} \\
  2&{ - 6}
\end{array}}} = \dfrac{1}{{\begin{array}{*{20}{c}}
  3&{ - 1} \\
  2&1
\end{array}}}$
$ \Rightarrow \dfrac{x}{{6 + 4}} = \dfrac{{ - y}}{{ - 18 + 8}} = \dfrac{1}{{3 + 2}}$
$ \Rightarrow \dfrac{x}{{10}} = \dfrac{{ - y}}{{ - 10}} = \dfrac{1}{5}$
$ \Rightarrow x = \dfrac{{10}}{5} = 2$ and $y = \dfrac{{10}}{5}=2$
Hence, the solution of the given system of equations is $x = 2$, $y = 2$.