Question

How do you solve the simultaneous equations $y = 3x$ and $x + 2y = 21$?

Answer 1

Hint: In this question, we are asked to solve the two given equations. Since we are given an expression for the variable y as one of the equations, the simplest way to solve this pair of equations by the method of substitution. So we substitute the expression of y in the other equation and we solve for the variable x. Then we substitute the value of x in the expression of y and obtain the value of y. So we obtain both the values of the unknown variable by substitution.

Complete step by step answer:
Given the two equations of the form,
$y = 3x$ …… (1)
$x + 2y = 21$ …… (2)
If we carefully observe the above two equations, we note that the equation (1) is an expression for the variable y. So we try to solve the given problem using the method of substitution.
In this method we substitute the value of one variable from one equation in the other equation. So that the other equation gets converted into a linear equation of only one variable.
Now we use the substitution method to find the values of unknown variables x and y.
Substituting $y = 3x$ in the equation (2) we get,
$ \Rightarrow x + 2(3x) = 21$
$ \Rightarrow x + 6x = 21$
Combining the like terms we get,
$ \Rightarrow 7x = 21$
Dividing throughout the equation by 7 we get,
$ \Rightarrow \dfrac{{7x}}{7} = \dfrac{{21}}{7}$
$ \Rightarrow x = 3$.
Now we substitute the value of x in the equation (1) to obtain the value of y.
Substituting $x = 3$ in the equation (1), we have,
$ \Rightarrow y = 3 \times 3$
$ \Rightarrow y = 9$.
Hence we have $x = 3$ and $y = 9$.

Therefore, the solution for the simultaneous equations $y = 3x$ and $x + 2y = 21$ is given by
$x = 3$ and $y = 9$.

Note: The substitution method is the algebraic method to solve the simultaneous equations. As the word says in this method the value of one variable from one equation is substituted in the other equation. In this way a pair of the linear equations gets transformed into one linear equation with only one variable, which can be easily solved.
Also note that we can verify the obtained values of x and y by substituting back in any one of the given equations.
In the above problem we have obtained $x = 3$ and $y = 9$. We substitute in $x + 2y = 21$ we get,
$ \Rightarrow 3 + 2(9) = 21$
$ \Rightarrow 21 = 21$
Note that L.H.S. is equal to the R.H.S. Hence the obtained values of x and y are correct.