Question

By substitution method 4m+6n=54; 3m+2n=28

Answer 1

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$4m + 6n = 54.......a \\ 3m + 2n = 28.......b \\ eq. \: a \times 1 - eq. \: b \times 3 \\ 4m + 6n = 54 \\ - 9m - 6n = - 84 \\ - 5m = 30 \\ m = - 6 \\ \\ put \: \: the \: \: value \: \: of \: \: m \: \: in \: eq. \: \: a \\ 4( - 6) + 6n = 54 \\ - 24 + 6n = 54 \\ 6n = 54 + 24 \\ 6n = 78 \\ n = 13$

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Answer 2

Answer:

The solution of the given simultaneous equations is ( m, n ) = ( 6, 5 ).

Step-by-step-explanation:

The given simultaneous equations are 4m + 6n = 54 and 3m + 2n = 28.

$\sf\:4m\:+\:6n\:=\:54\\\\\\\implies\sf\:4m\:=\:54\:-\:6n\\\\\\\implies\sf\:m\:=\:\dfrac{54\:-\:6n}{4}\\\\\\\implies\sf\:m\:=\:\dfrac{-\:6n\:+\:54}{4}\:\:\:-\:-\:(\:1\:)\\\\\\\sf\:3m\:+\:2n\:=\:28\\\\\\\implies\sf\:3\:\times\:\bigg(\:\dfrac{-\:6n\:+\:54}{4}\:\bigg)\:+\:2n\:=\:28\:\:\:-\:-\:[\:From\:(\:1\:)\\\\\\\implies\sf\:\dfrac{-\:18n\:+\:162}{4}\:+\:2n\:=\:28\\\\\\\implies\sf\:-\:18n\:+\:162\:+\:8n\:=\:112\:\:\:-\:-\:[\:Multiplying\:by\:4\:]\\\\\\\implies\sf\:-\:10n\:+\:162\:=\:112\\\\\\\implies\sf\:-\:10n\:=\:112\:-\:162\\\\\\\implies\sf\:\cancel{-}\:10n\:=\:\cancel{-}\:50\\\\\\\implies\sf\:n\:=\:\cancel{\frac{50}{10}}\\\\\\\implies\boxed{\red{\sf\:n\:=\:5}}$

By substituting n = 5 in equation ( 1 ), we get,

$\sf\:m\:=\:\dfrac{-\:6n\:+\:54}{4}\\\\\\\implies\sf\:m\:=\:\dfrac{-\:6\:\times\:5\:+\:54}{4}\\\\\\\implies\sf\:m\:=\:\dfrac{-\:30\:+\:54}{4}\\\\\\\implies\sf\:m\:=\:\cancel{\frac{24}{4}}\\\\\\\implies\boxed{\red{\sf\:m\:=\:6}}$

Additional Information:

1. Linear Equations in two variables:

The equation with the highest index (degree) 1 is called as linear equation. If the equation has two different variables, it is called as 'linear equation in two variables'.

The general formula of linear equation in two variables is

ax + by + c = 0

Where, a, b, c are real numbers and

a ≠ 0, b ≠ 0.

2. Solution of a Linear Equation:

The value of the given variable in the given linear equation is called the solution of the linear equation.