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

seo-qna
Last updated date: 25th Jun 2024
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Answer
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Hint: Where we are given a pair of linear equations and we have to find the value of x and y by using the given equation. We can solve the equation by the method of elimination or by using the method of substitution for the method of substitution. First we will find the value of one variable in the form of another for example we will find the value of x in terms of y then substitute that value in another equation. Then we will solve the equation and find the value of that variable. After that substitute the value of that variable in another equation and find the value of the remaining one variable.

Complete step-by-step answer:
Step1: We are given a pair of linear equations $3x - 2y = 7$ and $x + 3y = - 5$ by applying the method of substitution we will find the value of both variables. We will solve the second equation for $x$:
$ \Rightarrow x + 3y = - 5$
Subtracting $3y$ from both sides:
$ \Rightarrow x + 3y - 3y = - 5 - 3y$
On proper rearrangement we will get:
$ \Rightarrow x = - 5 - 3y$
Step2: Now we will substitute the value of x in the first equation and solve for $y$:
$ \Rightarrow 3( - 5 - 3y) - 2y = 7$
$ \Rightarrow - 15 - 9y - 2y = 7$
Adding $15$ both the sides we will get:
$ \Rightarrow 15 - 15 - 11y = 15 + 7$
$ \Rightarrow 0 - 11y = 22$
Dividing both sides by$ - 11$:
$ \Rightarrow \dfrac{{ - 11y}}{{ - 11}} = \dfrac{{22}}{{ - 11}}$
$ \Rightarrow y = - 2$
Step3: Substitute $ - 2$ for y in the solution to the second equation at the end of step1 and calculate $x$:
$ \Rightarrow x = - 5 - \left( {3 \times - 2} \right)$
On further solving we will get:
$ \Rightarrow x = - 5 + 6$
$x = 1$
So the solution is $x = 1;y = - 2$

Hence the solution is $x = 1;y = - 2$

Note:
This type of question we can solve by two methods: first is substitution and the second one is elimination. In this method the main thing is to find the value of one variable in terms of other students mainly doing the mistakes here.
Alternate method:
We are given two equations i.e.
$3x - 2y = 7$…(1)
$x + 3y = - 5$….(2)
Multiply (2) equation by $3$
$3x + 9y = - 15$…(3)
Subtract equation (3) from (1)
$ 3x+ 9y = - 15 $
$ \underline {( - ){{3x}} - 2y = 7} $
$ 11y = - 22 $
Now dividing the both sides by $11$
$y = - 2$
Substitute $y = - 2$ in equation (1) we get the value of x
$ \Rightarrow 3x + 4 = 7$
$ \Rightarrow 3x = 3$
$ \Rightarrow x = 1$
Here also we will get the same solution i.e. $x = 1;y = - 2$