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If \[3m + 5n = 9\] and \[5m + 3n = 7\] then find the value of \[8m + 8n\].

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Last updated date: 25th Jun 2024
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Answer
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Hint:
Here we will multiply both the given equation by a suitable number such that the coefficient of either of the two variables will become the same. Then we will use the elimination method to find the value of one of the variables. Then we will substitute the value of the obtained variable in one of the equations to find the other variable. We will substitute the value of the variables in the given algebraic expression to get its value.

Complete step by step solution:
Here we need to find the value of given algebraic expression i.e. \[8m + 8n\].
We have been given the two algebraic equations and they are:-
\[3m + 5n = 9\] ………….. \[\left( 1 \right)\]
\[5m + 3n = 7\] ………….. \[\left( 2 \right)\]
Now, we will multiply the equation \[\left( 1 \right)\] by 5. Therefore, we get
\[15m + 25n = 45\]………………………..\[\left( 3 \right)\]
We will now multiply the equation \[\left( 2 \right)\] by 3. Therefore, we get
\[15m + 9n = 21\]…………………………….\[\left( 4 \right)\]
Now subtracting equation \[\left( 4 \right)\] from the equation \[\left( 3 \right)\], we get
\[\begin{array}{l}15m + 25n - \left( {15m + 9n} \right) = 45 - 21\\ \Rightarrow 15m + 25n - 15m - 9n = 24\end{array}\]
Adding and subtracting the like terms, we get
\[ \Rightarrow 0 + 16n = 24\]
Now, we will divide both sides by 16.
\[\begin{array}{l} \Rightarrow \dfrac{{16n}}{{16}} = \dfrac{{24}}{{16}}\\ \Rightarrow n = \dfrac{3}{2}\end{array}\]
Now, we will substitute the value of the \[n\] in equation \[\left( 1 \right)\].
 \[3m + 5 \times \dfrac{3}{2} = 9\]
On multiplying the terms, we get
\[ \Rightarrow 3m + \dfrac{{15}}{2} = 9\]
Now, we will subtract \[\dfrac{{15}}{2}\] from both sides. Therefore, we get
\[\begin{array}{l} \Rightarrow 3m + \dfrac{{15}}{2} - \dfrac{{15}}{2} = 9 - \dfrac{{15}}{2}\\ \Rightarrow 3m = \dfrac{{18 - 15}}{2}\end{array}\]
Subtracting the terms in the numerator, we get
\[ \Rightarrow 3m = \dfrac{3}{2}\]
Now, we will divide both sides by 3.
\[ \Rightarrow \dfrac{{3m}}{3} = \dfrac{1}{3} \times \dfrac{3}{2}\]
On further simplification, we get
\[ \Rightarrow m = \dfrac{1}{2}\]
But we need to find the value of \[8m + 8n\].
So, we will substitute the value of both the variables in the above expression.
\[8m + 8n = 8 \times \dfrac{1}{2} + 8 \times \dfrac{3}{2}\]
Simplifying the expression, we get
\[ \Rightarrow 8m + 8n = 4 + 4 \times 3\]
On multiplying the numbers, we get
\[ \Rightarrow 8m + 8n = 4 + 12\]
On adding the numbers, we get
\[ \Rightarrow 8m + 8n = 16\]

Hence, the value of the given expression \[8m + 8n\] is equal to 16.

Note:
Here we have obtained the value of the given algebraic expression. An algebraic expression in mathematics is defined as an expression which is made up of constants and variables, along with algebraic operations like subtraction, addition and division etc.
There is an alternate method to solve this problem.
To find the value of the given expression i.e. \[8m + 8n\], we will add the equation\[\left( 1 \right)\] and equation \[\left( 2 \right)\].
\[\begin{array}{l}3m + 5n + 5m + 3n = 9\\ \Rightarrow 8m + 8n = 16\end{array}\]
Hence, the value of the given expression \[8m + 8n\] is equal to 16.