Question

How do you solve $x + 5y - 10 = 0$ and $x = 2y - 8$ using substitution?

Answer 1

Hint: First of all, using the value of x from the second equation to put it in the first equation will lead us to the value of y. Substituting that back in equation number 2, the value of x can be found as well.

Complete step-by-step solution:
We are given that we are required to solve $x + 5y - 10 = 0$ and $x = 2y - 8$ using substitution.
Let us term the given equation $x + 5y = 10$ as the equation number 1 and the given equation $x = 2y - 8$ as equation number 2.
Putting equation number 2 in equation number 1, we will then obtain the following expression with us:-
$ \Rightarrow \left( {2y - 8} \right) + 5y = 10$
Simplifying the left hand side of the above expression, we will then obtain the following equation:-
$ \Rightarrow 2y - 8 + 5y = 10$
Simplifying the left hand side of the above expression further, we will then obtain the following equation:-
$ \Rightarrow 7y - 8 = 10$
Taking 8 from subtraction in the left hand side of the above expression to addition in the right hand side, we will then obtain the following expression:-
$ \Rightarrow 7y = 18$
Simplifying this further, we will then obtain the following equation:-
$ \Rightarrow y = \dfrac{{18}}{7}$
Thus, we get: $y = \dfrac{{18}}{7}$
Putting this in equation number 2, we will then obtain the following equation:-
$ \Rightarrow x = 2\left( {\dfrac{{18}}{7}} \right) - 8$
Simplifying the calculations, we will then obtain the following equation:-
$ \Rightarrow x = - \dfrac{{20}}{7}$
Hence, the answer is $x = - \dfrac{{20}}{7}$ and $y = \dfrac{{18}}{7}$.

Note: The students must note that you may use alternate methods for solving the equations other than using the substitution method if not mentioned in the question.
Alternate Way:
We are given that we are required to solve $x + 5y = 10$ …………(1) and $x = 2y - 8$ ………(2)
Multiplying the equation 1 by 2 and 2 by 5, we will then obtain the following equations respectively:
$ \Rightarrow 2x + 10y = 20$ ……………(3)
$ \Rightarrow 5x = 10y - 40$ ……………(4)
Adding the equation number 4 and equation number 3, we will then obtain the following equation:-
$ \Rightarrow \left\{ {2x + 10y} \right\} + \left\{ {5x} \right\} = 20 + 10y - 40$
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow 7x = - 20$
Thus, we have $x = - \dfrac{{20}}{7}$
Therefore, by putting this in equation number 2, we get $y = \dfrac{{18}}{7}$.
Hence, the answer is $x = - \dfrac{{20}}{7}$ and $y = \dfrac{{18}}{7}$.