Question

How do you solve the system $y = 2x$ and $x + 3y = - 14$ using substitution?

Answer 1

Hint: We need to solve this question using a substitution method. We are provided with two equations, one has a single variable while the other equation has two variables. We substitute the value provided for $y$ in the second equation so that it becomes saturated and has only one variable. Then we solve it further to get the value of $x$. After we have found the value of $x$, we substitute it in $y = 2x$, to find our required answer.

Complete step-by-step solution:
We are given two sets of equations for two lines in this question. The given equations are $y = 2x$ and $x + 3y = - 14$, now we need to solve this in order to get the value of the variables $x$ and $y$.
We use a substitution method to solve this sum. We are provided with two equations here: the first equation contains only the variable $y$ as a constraint while the other equation contains both the variables.
In substitution method, we simply place the constraint provided for the variable $y$ in the second equation and solve for $x$
$y = 2x$ → First equation
$x + 3y = - 14$ → Second equation
On substituting the value of $y$ in the second equation, we get:
$ \Rightarrow x + 3\left( {2x} \right) = - 14$
Now we have a single variable in this equation, therefore we can easily solve it to find the value of $x$
$ \Rightarrow x + 6x = - 14$
On solving it further, we get:
$ \Rightarrow 7x = - 14$
On dividing both sides of the equation with $7$, we get:
$ \Rightarrow x = - 2$
Now, according to the first equation:
$ \Rightarrow y = 2x$
Therefore, $y = 2\left( { - 2} \right) = - 4$

The coordinates are $\left( { - 2, - 4} \right)$

Note: When two sets of equations are given, then we can either solve them simultaneously or using the substitution method. The latter is the easiest method to be applied and can be applied only if we have an equation containing the value or constraints for a single variable. Once we place this value or constraint in the other equation containing two variables, we can easily solve.