Question

How do you solve $\dfrac{1}{2}x + \dfrac{1}{3}y = 5$ and $\dfrac{1}{4}x + y = 10$ using substitution?

Answer 1

Hint: This problem deals with solving two equations with two variables. Given a pair of linear equations with variables $x$ and $y$. Now we have to solve these two linear equations with two variables, with the help of a substitution method. If given two variables, then we know that we need two equations to solve these two variables.

Complete step-by-step solution:
The given two linear equations are $\dfrac{1}{2}x + \dfrac{1}{3}y = 5$ and $\dfrac{1}{4}x + y = 10$
Consider the first equation, as given below:
$ \Rightarrow \;\dfrac{1}{2}x + \dfrac{1}{3}y = 5$
Now simplify the left hand side of the above equation, as given below:
$ \Rightarrow \;\dfrac{{3x + 2y}}{6} = 5$
Now multiply the above equation with 6 on both sides of the equation, as shown below:
$ \Rightarrow \;3x + 2y = 5\left( 6 \right)$
$ \Rightarrow \;3x + 2y = 30$
So the simplified first equation is $\;3x + 2y = 30$.
Now consider the second equation, as shown below:
$ \Rightarrow \dfrac{1}{4}x + y = 10$
Now simplify the left hand side of the above equation, as given below:
$ \Rightarrow \dfrac{{x + 4y}}{4} = 10$
Now multiply the above equation with 6 on both sides of the equation, as shown below:
$ \Rightarrow x + 4y = 10\left( 4 \right)$
$ \Rightarrow x + 4y = 40$
So the simplified second equation is $x + 4y = 40$.
Now consider the second equation, as given below:
$ \Rightarrow x + 4y = 40$
From here get the expression of $x$, as given below:
$ \Rightarrow x = 40 - 4y$
Now substitute the obtained expression of $x$ in the first equation as shown below:
We know that the first equation is $\;3x + 2y = 30$,
$ \Rightarrow \;3\left( {40 - 4y} \right) + 2y = 30$
Now simplify the left hand side of the equation:
$ \Rightarrow \;120 - 12y + 2y = 30$
$ \Rightarrow \;120 - 10y = 30$
Now grouping the constants on the other side of the equation:
$ \Rightarrow \;10y = 120 - 30$
$ \Rightarrow \;10y = 90$
Now divide the equation by 10, as shown:
$\therefore y = 9$
Now substitute the value of $y = 9$ in the second equation:
$ \Rightarrow x + 4\left( 9 \right) = 40$
$ \Rightarrow x = 40 - 36$
$\therefore x = 4$

The values of x and y are 4 and 9 respectively.

Note: Please note that the given two linear equations with two variables are solved by substitution method, to verify the solutions obtained, we can solve the given two linear equations by the elimination method, which is by adding the two equations, when matched the coefficients of x or y, and then check whether the obtained are correct.