Question

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

Answer 1

Hint: In this question, we can solve the given equation and then we find the value of $x$ and $y$ , by using the substitution method. This equation deals with linear equations with two variables.
The idea here is to solve one of the equations for one of the variables, and put this into the other equation. It doesn't matter whichever equation or variable you pick. There is no right or wrong choice of equation; the answer will be the same.
We already know that these equations represent the same line, that is, this is a dependent system. We get two identical line equations.

Complete step-by-step solution:
Mark the given two equations as $\left( 1 \right)$ and $\left( 2 \right)$
$y - 3x = 7$ …. $\left( 1 \right)$
$21 + 9x = 3y$ …. $\left( 2 \right)$
Equation $\left( 1 \right)$ can be written in the following form
$y = 7 + 3x$
Now, by using the substitution method we can substitute $y = 7 + 3x$ in equation $\left( 2 \right)$ and we will get the value of $x$ .
$21 + 9x = 3y$
Putting $y = 7 + 3x$ ,
$21 + 9x = 3\left( {7 + 3x} \right)$
Multiplying inside the brackets,
$21 + 9x = 21 + 9x$
This implies that $x$ can be any real number as the equation is true for all real numbers.
I substituted the first equation into the second equation, so this unhelpful result is not because of my choice of equations. It’s just that this is what a dependent system looks like when you try to find a solution. We are trying to find the one single point that works in both equations.
Equation $\left( 1 \right)$ and $\left( 2 \right)$ are the same line. The lines intersect at infinitely many points.
In other words, I got an unhelpful result because the lines were overlapping. This tells me that the system is actually dependent and that the solution is the whole line.
$y = 7 + 3x$
And the solution is $\left( {x,7 + 3x} \right)$ .
The solution is the line $y = 7 + 3x$

Note: A system of linear equations is just a set of two or more linear equations.
In two variables, the graph of a system of two equations is a pair of lines in the plane.
There are three possibilities:
The lines intersect at zero points. (The lines are parallel.)
The lines intersect at exactly one point. (The lines are intersecting.)
The lines intersect at infinitely many points. (The lines are overlapping.)