Question

Solve: $2x + y + 13 = 0$ and $2x - 3y = 9$.

Answer 1

Hint:In the given question, we need to solve two simultaneous equations in two variables. There are various methods to solve two given equations in two variables like substitution method, cross multiplication method, elimination method, matrix method and many more. The equations given in the question can be solved using any one of the above mentioned methods easily. But we will solve the equations using the substitution method. We will substitute the value of variable y from the first equation into the second one and find the value of both variables.

Complete step by step answer:
In the question, we are given a couple of simultaneous linear equation in two variables.
$2x + y + 13 = 0$
$2x - 3y = 9$
In substitution method, we substitute the value of one variable from an equation into other equation so as to get an equation in only one variable.
So, we have, $2x + y + 13 = 0$
Isolating the variable y by shifting all the remaining terms to right side of equation, we get,
$ \Rightarrow y = - \left( {2x + 13} \right)$
Now, we substitute the value of y in to the second equation. So, we get,
\[2x - 3\left( { - \left( {2x + 13} \right)} \right) = 9\]
Opening the brackets,
\[ \Rightarrow 2x + 6x + 39 = 9\]
Shifting all constants to right side,
\[ \Rightarrow 8x = 9 - 39\]
Simplifying the calculations and dividing both sides by $8$.
\[ \Rightarrow x = \dfrac{{ - 30}}{8}\]
Cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow x = \dfrac{{ - 15}}{4}\]
So, we get the value of x as \[\left( { - \dfrac{{15}}{4}} \right)\].
Putting the value of x in any of the two equations, we get,
$2x + y + 13 = 0$
$ \Rightarrow 2\left( { - \dfrac{{15}}{4}} \right) + y + 13 = 0$
$ \Rightarrow - \dfrac{{15}}{2} + y + 13 = 0$
Shifting constants to right side,
$ \Rightarrow y = \dfrac{{15}}{2} - 13$
Taking LCM of rational expressions,
$ \Rightarrow y = \dfrac{{15 - 26}}{2}$
$ \Rightarrow y = \dfrac{{ - 11}}{2}$
So, the value of y is $\dfrac{{ - 11}}{2}$.

Therefore, the solution of the simultaneous linear equations $2x + y + 13 = 0$ and $2x - 3y = 9$ is $x = \left( { - \dfrac{{15}}{4}} \right)$ and $y = \left( { - \dfrac{{11}}{2}} \right)$.

Note:An equation consisting of 2 variables having degree one is known as Linear Equation in two variables. Standard form of Linear Equation in two variables is $ax + by + c = 0$ where a, b and c are the real numbers and a, b which are coefficients of x and y respectively are not equal to 0. We must take care of the calculations while substituting the variables from one equation into another equation.