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Hint: We have been given two equations. Write the equations in the form of $ax+by+c=0$. Then use $\dfrac{m}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\dfrac{n}{{{c}_{1}}{{a}_{2}}-{{a}_{1}}{{c}_{2}}}=\dfrac{-1}{{{a}_{2}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$. You will get the answer.

Complete step-by-step answer:

The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.

Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have given lengths. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis.

Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation. The exercise set will probably start out by asking for the solutions to straight forward simple proportions, but they might use the "odds" notation.

Specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. Each step in these procedures is based on a single, fundamental property of equations. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to students.

The method is also occasionally known as the "cross your heart" method because a heart can be drawn to remember which things to multiply together and the lines resemble a heart outline.

In practice, though, it is easier to skip the steps and go straight to the "cross-multiplied" form.

So we have been given two equations.

$3m+n-15={{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$………… (1)

$m+2n-10={{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$……….. (2)

Now using the formula for cross multiplication,

$\dfrac{m}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\dfrac{n}{{{c}_{1}}{{a}_{2}}-{{a}_{1}}{{c}_{2}}}=\dfrac{-1}{{{a}_{2}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$

So from equation (1) and (2),

$\dfrac{m}{1\times (-10)-2\times (-15)}=\dfrac{n}{(-15)\times 1-3\times (-10)}=\dfrac{1}{3\times 2-1\times 1}$

$\dfrac{m}{-10+30}=\dfrac{n}{-15+30}=\dfrac{1}{6-1}$

Simplifying we get,

$\dfrac{m}{20}=\dfrac{n}{15}=\dfrac{1}{5}$

Now equating we get,

$\dfrac{m}{20}=\dfrac{1}{5}$

So simplifying we get,

$m=4$

Also, \[\dfrac{n}{15}=\dfrac{1}{5}\]

$n=3$

Here we get, $(m,n)=(4,3)$.

So the solution is $(4,3)$.

So the correct answer is option(A).

Note: Carefully read the question. Don’t be confused about the cross multiplication method. While simplifying, do not make mistakes. Don’t miss any term while solving. Take care that no terms are missing.

Complete step-by-step answer:

The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.

Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have given lengths. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis.

Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation. The exercise set will probably start out by asking for the solutions to straight forward simple proportions, but they might use the "odds" notation.

Specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. Each step in these procedures is based on a single, fundamental property of equations. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to students.

The method is also occasionally known as the "cross your heart" method because a heart can be drawn to remember which things to multiply together and the lines resemble a heart outline.

In practice, though, it is easier to skip the steps and go straight to the "cross-multiplied" form.

So we have been given two equations.

$3m+n-15={{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$………… (1)

$m+2n-10={{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$……….. (2)

Now using the formula for cross multiplication,

$\dfrac{m}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\dfrac{n}{{{c}_{1}}{{a}_{2}}-{{a}_{1}}{{c}_{2}}}=\dfrac{-1}{{{a}_{2}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$

So from equation (1) and (2),

$\dfrac{m}{1\times (-10)-2\times (-15)}=\dfrac{n}{(-15)\times 1-3\times (-10)}=\dfrac{1}{3\times 2-1\times 1}$

$\dfrac{m}{-10+30}=\dfrac{n}{-15+30}=\dfrac{1}{6-1}$

Simplifying we get,

$\dfrac{m}{20}=\dfrac{n}{15}=\dfrac{1}{5}$

Now equating we get,

$\dfrac{m}{20}=\dfrac{1}{5}$

So simplifying we get,

$m=4$

Also, \[\dfrac{n}{15}=\dfrac{1}{5}\]

$n=3$

Here we get, $(m,n)=(4,3)$.

So the solution is $(4,3)$.

So the correct answer is option(A).

Note: Carefully read the question. Don’t be confused about the cross multiplication method. While simplifying, do not make mistakes. Don’t miss any term while solving. Take care that no terms are missing.

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