Question

If $ - 2x = 4y + 6$ and $2\left( {2y + 3} \right) = 3x - 5$, what is the solution $\left( {x,y} \right)$ to the system of equations above?

Answer 1

Hint: In the above question, first we will write both the equations in such a manner that the coefficient of $x$ should be positive and there should be no bracket like that we have in the second equation. Then we will use the elimination method to simplify both the equations to find the value of $x$ and $y$.

Complete step by step answer:
In the above question, we have given two equations $ - 2x = 4y + 6$ and $2\left( {2y + 3} \right) = 3x - 5$. We have to solve them simultaneously to find the value of x and y.We have,
$ - 2x = 4y + 6$
Now, we will transpose $ - 2x$ to the right-hand side and $6$ to the left-hand side.
$ \Rightarrow 2x + 4y = - 6.....................\left( 1 \right)$
Also, we have
$2\left( {2y + 3} \right) = 3x - 5$
Now, multiplying on the left hand side.
$ \Rightarrow 4y + 6 = 3x - 5$
On transposing the values, we get
$ \Rightarrow 3x - 4y = 6 + 5$
$ \Rightarrow 3x - 4y = 11..............\left( 2 \right)$

Now adding both the equations
$ \Rightarrow 5x = 5$
Now, divide both sides by $5$.
$ \therefore x = 1$
Now, put the value of x in equation $\left( 1 \right)$.
$ \Rightarrow 2\left( 1 \right) + 4y = - 6$
Now transpose the value of two in the right-hand side.
$ \Rightarrow 4y = - 6 - 2$
$ \Rightarrow 4y = - 8$
Now, divide both sides by four.
$ \therefore y = - 2$

Therefore, the value of $\left( {x,y} \right)$ is $\left( {1, - 2} \right)$.

Note: Any equation which can be put in the form \[{\mathbf{ax}}{\text{ }} + {\text{ }}{\mathbf{by}}{\text{ }} + {\text{ }}{\mathbf{c}}{\text{ }} = {\text{ }}{\mathbf{0}}\], where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. This means that you can think of many such equations. We can also solve this equation by using the method of substitution in which we have to find the value of one variable in terms of another variable and then we will put its value in the second equation to find the value of the variable.