Question

How do you solve by substitution $y – 3x = 7$ and $21 + 9x = 3y$?

Answer 1

Hint: We will first find the value of $y$ from the given first equation and then put in this value of $y$ in terms of $x$ in the second equation to get the value of $x$.

Complete step-by-step solution:
We are given that we are required to solve $y – 3x = 7$ and $21 + 9x = 3y$ by substitution.
Let us assume the equation as $y – 3x = 7$ ………………(1) (equation number 1)
And $21 + 9x = 3y$ ……………..(2) (equation number 2)
Now, we have two equations.
Considering first equation: $y – 3x = 7$
Taking 3x from subtraction in the left hand side to addition in right hand side, we will get:-
$ \Rightarrow $$y = 3x + 7$
Now, putting this value of y in equation number (2), we will then get:-
$ \Rightarrow $$21 + 9x = 3(3x + 7)$
$ \Rightarrow $$21 + 9x = 9x + 21$
Since both sides are equal, this means that both the lines are coincident and this is true for all the values of x.
Thus this is true for all the values of y as well.

Note: The students must note that if not mentioned that you have to solve this by substitution. Then we also have alternate ways:
Alternate way 1:
We will multiply the equation number 1 by 3 on both the sides to obtain the following equation:-
$ \Rightarrow $$3y – 9x = 21$
Now, adding this to equation number 2 which is given by 21 + 9x = 3y, we will then obtain the following equation:-
$ \Rightarrow $$3y – 9x + 21 + 9x = 21 + 3y$
$ \Rightarrow $0 = 0
Thus, it is true for all the values of x and y.
Alternate way 2:
You may also draw the graph of both the equations and the lines will come out to be coincident and thus infinite solutions.