Question

How do you solve the system of equations $2x + 3y = 7$ and $ - 3x - 5y = - 13$?

Answer 1

Hint: Here we have the given equations are linear; we will solve the equations by the method of elimination by getting similar coefficients in a variable to reduce it. Finally we get the required answer.

Complete step-by-step solution:
We have the given equations as:
$2x + 3y = 7 \to (1)$
$ - 3x - 5y = - 13 \to (2)$
Now to get the coefficients similar we will multiply equation $(1)$ with $3$.
On multiplying we get:
$ \Rightarrow 6x + 9y = 21 \to (3)$
And we will multiply equation $(2)$ with $2$.
On multiplying we get:
$ \Rightarrow - 6x - 10y = - 26 \to (4)$
Now on adding equation $(3)$ and $(4)$ we get:
$ \Rightarrow - y = - 5$
On rearranging the position of the terms, we can write the equation as:
$y = 5$, which is the value of $y$.
Now to get the value of $x$ we will substitute the value of $y = 5$ in equation $(1)$.
On substituting we get:
$ \Rightarrow 2x + 3(5) = 7$
On simplifying we get:
$ \Rightarrow 2x + 15 = 7$
On sending $15$ across the $ = $ sign, we get:
$ \Rightarrow 2x = - 8$
Therefore, on simplifying we get:
 $x = - 4$, which is the value of $x$.

Therefore, the value of $x$ is $ - 4$ and the value of $y$ is $5$.

Note: To check whether the solution is correct we have to test the values of $x$ and $y$ in equation $(2)$
On substituting the values in the left-hand side of the equation, we get:
$ \Rightarrow - 3( - 4) - 5(5)$
On simplifying we get:
\[ \Rightarrow 12 - 25\]
Which is equal to $13$, which is the right-hand side, therefore the solution is correct.
It is to be remembered that in any given equation multiplying or dividing the equation by a specific constant doesn’t change the value of the equation.
In the given question we had two variables which are $x$ and $y$, therefore they can be solved by using elimination, where there are more than three variables; matrix is used to solve them.