Question

m$\left[\begin{array}{ccc}-3&4\\\end{array}\right] +n\left[\begin{array}{ccc}4&-3\\\end{array}\right] =\left[\begin{array}{ccc}10&-11\\\end{array}\right]$ find the value of 3m+7n.

Answer 1

✿ $\textbf{\textsf{\large \underline{\red{Solution}}:-}}$

● Firstly we need to find the value of m & n in ,

$\sf\longrightarrow m\big[ \begin{array}{cc}\sf -3 & \sf 4\end{array}\big] + n\big[\begin{array}{cc}\sf 4 & \sf -3\end{array}\big] = \big[\begin{array}{cc}\sf 10&\sf -11\end{array}\big]$

Taking LHS & multiplying ,

$\sf\longrightarrow \big[\begin{array}{cc}\sf -3m &\sf 4m\end{array}\big]+\big[\begin{array}{cc}\sf 4n & \sf -3n\end{array}\big]$

Now , adding both the matrices & comparing with RHS

$\sf\longrightarrow \big[\begin{array}{cc}\sf -3m+4n &\sf 4m-3n\end{array}\big]=\big[\begin{array}{cc}\sf 10&\sf -11\end{array}\big]$

Now , by comparing

⊙ - 3m + 4n = 10 ….eq ( 1 )

⊙ 4m - 3n = - 11 ….eq ( 2 )

Simplifying eq ( 1 )

$\to$ - 3m + 4n = 10

Multiply with 3 on both sides

$\to$ 3( - 3m + 4n ) = 3(10)

$\to$ -9m + 12n = 30 ….eq ( 3 )

Simplifying eq ( 4 )

$\to$ 4m - 3n = - 11

Multiplying with 4 on both sides

$\to$ 4(4m - 3n) = 4(-11)

$\to$ 16m - 12n = - 44 ....eq ( 4 )

Now , eq ( 3 + 4)

$\leadsto$ - 9m + 12n + [ 16m - 12n ] = 30 + ( - 44)

$\leadsto$ - 9m + 12n + 16m - 12n = 30 - 44

$\leadsto$ 7m = - 14

$\leadsto$ m = -14 ÷ 7

$\leadsto$ m = -2

Now , substituting the value of m in eq ( 1 )

$\mapsto$ - 3(-2) + 4n = 10

$\mapsto$ 6 + 4n = 10

$\mapsto$ 4n = 10 - 6

$\mapsto$ n = $\sf\dfrac{4}{4}$

$\mapsto$ n = 1

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Now , finding 3m + 7n

Substituting the values of m & n

$\hookrightarrow$ 3( - 2 ) + 7(1)

$\hookrightarrow$ - 6 + 7

$\hookrightarrow$ 3m + 7n = 1

Hence , 3m + 7n = 1