Question

Solve the following simultaneous equations: $4m + 6n = 54$ and $3m + 2n = 28$

Answer 1

Hint: Here we are asked to solve for the value of $m$ and $n$ by using the given two linear equations. Any linear set of equations can be solved by using the substitution method. In this method, we will first find the expression for any one of the unknown variables by rearranging any one of the given equations; this expression will act as a value for that unknown variable. Then we will substitute it in any one of the equations to make it an equation with only one variable which will be easier to solve and find the value of that variable then it can be substituted in any one of the equations to find the value of another variable.

Complete step by step solution:
Given that $4m + 6n = 54$ and $3m + 2n = 28$ mark it as the equation $1,2$ respectively.
Now let us rearrange the equation $2$ to get an expression for the unknown variable $m$
$3m + 2n = 28 $
$\Rightarrow m = \dfrac{{28 - 2n}}{3}$
Now let us apply the values in equation
$1$ we get, $4m + 6n = 54 $
$\Rightarrow 4(\dfrac{{28 - 2n}}{3}) + 6n = 54$
Further solving by cross multiplication, we get,
$4(28 - 2n) + 18n = 162$
Hence, we have by the multiplication,
$112 - 8n + 18n = 162$
By addition and subtraction, we get
$10n = 162 - 112 $
$\Rightarrow 10n = 50 $
$\Rightarrow n = 5$ (by division operation)
Now substitute the value in the equation one, we get
$4m + 6n = 54 $
$\Rightarrow 4m + 6(5) = 54 $
$\Rightarrow 4m = 54 - 30$
hence we get $4m = 24 \Rightarrow m = 6$
Hence, we got the two values as $m = 6,n = 5$

Note: We know that there are three methods to solve the linear equations with two unknown variables- the substitution method, the elimination method, and the matrix method. Other than the substitution method the elimination method is the one of the easiest methods. In the elimination method, we will try to eliminate any one of the unknown variables so that we will get an equation with only one unknown variable which will be easier to solve and then by substituting its value in any one of the given equations we will be able to find the value of another variable.