Question

How do you solve the pair of equations \[\dfrac{1}{2}x + \dfrac{1}{3}y = 2\] and \[\dfrac{1}{{3x}} + \dfrac{1}{{2y}} = \dfrac{{13}}{6}\] by reducing them to a pair of linear equations?

Answer 1

Hint: To solve the pair of given linear equations, combine all the like terms or by using any of the elementary arithmetic functions i.e., addition, subtraction, multiplication and division hence simplify the terms to get the value of \[x\] also the value of \[y\] by reducing them to a pair of linear equations.

Complete step by step solution:
 Given pair of equations are,
\[\dfrac{1}{2}x + \dfrac{1}{3}y = 2\] and \[\dfrac{1}{{3x}} + \dfrac{1}{{2y}} = \dfrac{{13}}{6}\]
\[\dfrac{1}{{2x}} + \dfrac{1}{{3y}} = 2\] …………… 1
\[\dfrac{1}{{3x}} + \dfrac{1}{{2y}} = \dfrac{{13}}{6}\] ……………. 2
Let,
\[\dfrac{1}{x} = u\] and \[\dfrac{1}{y} = v\]
Hence, we get equation 1 and equation 2 as:
\[\dfrac{1}{2}u + \dfrac{1}{3}v = 2\] ……………… 3
\[\dfrac{1}{3}u + \dfrac{1}{2}v = \dfrac{{13}}{6}\] ………………. 4
Now, Simplifying the terms of equation 3 we get
\[\dfrac{1}{2}u + \dfrac{1}{3}v = 2\]
\[ \Rightarrow \]\[\dfrac{{3u + 2v}}{{2 \times 3}} = 2\]
\[ \Rightarrow \]\[3u + 2v = 12\]
\[\dfrac{1}{3}u + \dfrac{1}{2}v = \dfrac{{13}}{6}\]
Simplifying the terms of equation 4 we get
\[ \Rightarrow \]\[\dfrac{{2u + 3v}}{{2 \times 3}} = \dfrac{{13}}{6}\]
\[ \Rightarrow \]\[2u + 3v = 13\]
Hence, the equations are:
\[3u + 2v = 12\] ………….. 5
\[2u + 3v = 13\] ……………. 6
From equation 5,
\[3u + 2v = 12\]
\[ \Rightarrow \]\[3u = 12 - 2v\]
\[ \Rightarrow \]\[u = \dfrac{{12 - 2v}}{3}\]
Now substitute the obtained value of u in equation 6 as,
\[2u + 3v = 13\]
\[ \Rightarrow \]\[2\left( {\dfrac{{12 - 2v}}{3}} \right) + 3v = 13\]
Multiplying both sides by 3
\[3 \times 2\left( {\dfrac{{12 - 2v}}{3}} \right) + 3v \times 3 = 13 \times 3\]
Simplify the terms:
\[2\left( {12 - 2v} \right) + 9v = 39\]
\[24 - 4v + 9v = 39\]
\[ - 4v + 9v = 39 - 24\]
\[5v = 15\]
\[v = \dfrac{{15}}{5} = 3\]
Now, substitute the value of v in equation 5 as:
\[3u + 2v = 12\]
\[3u + 2\left( 3 \right) = 12\]
\[3u + 6 = 12\]
\[3u = 12 - 6\]
\[u = \dfrac{6}{3} = 2\]
Hence, the value of u and v is:
\[u = 2\] and \[v = 3\].
Now, we need to find the value of x and y.
We know that
\[\dfrac{1}{x} = u\]
\[ \Rightarrow \]\[\dfrac{1}{x} = 2\]
\[ \Rightarrow \]\[x = \dfrac{1}{2}\]
And
\[\dfrac{1}{y} = v\]
\[ \Rightarrow \]\[\dfrac{1}{y} = 3\]
\[ \Rightarrow \]\[y = \dfrac{1}{3}\].
Hence, the value of x and y we obtained is:
\[x = \dfrac{1}{2}\] and \[y = \dfrac{1}{3}\]

Therefore, this is the solution of the given equation i.e, \[x = \dfrac{1}{2}\] and \[y = \dfrac{1}{3}\]

Note: The key point to solve these equations, is that we must arrange the terms such that to find the value of x and y we need to consider the terms of the given equations separately such that we can simplify the terms easily. We know that Simultaneous equations are two equations, each with the same two unknowns and are simultaneous because they are solved together, hence the key point to solve these kinds of equations we need to combine all the terms and then simplify the terms to get the value of x and y.