Question

How do solve the following linear system: $x+y=-1,4x-3y=1$?

Answer 1

Hint: To solve the given linear system $x+y=-1,4x-3y=1$ by using the substitution method, we will have to follow some steps that are:
First of all, we will rearrange one equation for $y$ in terms of $x$.
After that we will replace $y$ in another equation and solve the equation for getting the value $x$ .
Then for getting the value of $y$ , we will apply the value of $x$ in the equation that we already rearrange for $y$ in terms of $x$ .

Complete step by step solution:
Since, the given linear equation are:
$\Rightarrow x+y=-1$ … $\left( i \right)$
And
$\Rightarrow 4x-3y=1$ … $\left( ii \right)$
Let’s apply the substitution method to solve the given equation.
Firstly, we will rearrange the first equation for $y$ in terms of $x$ by changing the place of $x$ as:
$\Rightarrow y=-1-x$ … $\left( iii \right)$
Here, we can substitute $\left( -1-x \right)$ for $y$ with the help of equation $\left( iii \right)$ in equation $\left( ii \right)$ that is $4x-3y=1$ for finding the value of $x$ as:
$\Rightarrow 4x-3\left( -1-x \right)=1$
Now, we will open the bracket from the above equation as:
$\Rightarrow 4x-3\times \left( -1 \right)-\left( -3 \right)x=1$
Let’s find the solution for multiplication by multiplying $\left( -3 \right)$to $\left( -1-x \right)$as:
$\Rightarrow 4x+3+3x=1$
Since, $4x$ and $3x$ are equal like terms, we can combine them where we will add them according to the above equation as:
$\Rightarrow \left( 4x+3x \right)+3=1$
Here, we will get $7x$ by adding $4x$ and $3x$ as:
$\Rightarrow 7x+3=1$
Now, we will simplify the above equation. Let’s change place of numbers and place both the numbers one side as:
$\Rightarrow 7x=1-3$
Let’s continue the process for above equation as:
 $\Rightarrow 7x=-2$
From the above equation, we will have the value of $x$ that is:
$\Rightarrow x=\dfrac{-2}{7}$
Now, we will use this value of $x$ in equation $\left( iii \right)$ for getting the value of $y$ as:
$\Rightarrow y=-1-\left( -\dfrac{2}{7} \right)$
We know that the multiplication of two negative signs is a positive sign. So, the above equation will be as:
$\Rightarrow y=-1+\dfrac{2}{7}$
Since, both the signs of the number in the above equation are opposite; we will use subtraction method as:
 $\Rightarrow y=\left( \dfrac{2}{7}-1 \right)$
Here, we will convert rational number $1$ into fraction by multiplying $\dfrac{7}{7}$ as:
$\Rightarrow y=\left( \dfrac{2}{7}-1\times \dfrac{7}{7} \right)$
The above equation will be:
$\Rightarrow y=\left( \dfrac{2}{7}-\dfrac{7}{7} \right)$
Now, we will subtract $7$ and $4$ since the denominator is same for both numbers:
$\Rightarrow y=\left( \dfrac{2-7}{7} \right)$
After solving above equation, we will have the value of $y$as:
$\Rightarrow y=-\dfrac{5}{7}$
Hence, we got the values as $\left( -\dfrac{2}{7} \right)$ and $\left( -\dfrac{5}{7} \right)$ for $x$ and $y$ respectively by solving the given linear system by substitution method.

Note: Here, we will verify that the obtaining values for $x$ and $y$ are correct or not with help of the following method:
 In which we will put $\left( -\dfrac{2}{7} \right)$ and $\left( -\dfrac{5}{7} \right)$ in the place of $x$ and $y$ respectively in equation $\left( ii \right)$ where we will check if L.H.S. is equal to R.H.S or not.
The equation is $4x-3y=1$
Now, we will take L.H.S. that is:
$\Rightarrow 4x-3y$
Here, we will put $\left( -\dfrac{2}{7} \right)$ and $\left( -\dfrac{5}{7} \right)$ in the place of $x$ and $y$ respectively. So, the above equation will be:
$\Rightarrow 4\left( -\dfrac{2}{7} \right)-3\left( -\dfrac{5}{7} \right)$
After multiplying one negative and one positive sign, we will get the negative sign but when we will multiply the negative sign with negative sign, we will have a positive sign. So, the above equation we will be:
$\Rightarrow -\dfrac{4\times 2}{7}+\dfrac{3\times 5}{7}$
Now, we solve the calculation for the above equation. So we will get:
$\Rightarrow -\dfrac{8}{7}+\dfrac{15}{7}$
We can write above equation as:
$\Rightarrow \dfrac{15}{7}-\dfrac{8}{7}$
Since, the denominator is same in both terms, the above equation will get the value as:
$\Rightarrow \dfrac{7}{7}$
Now, we will find the final result from above term as:
$\Rightarrow 1$
That is equal to R.H.S.
Hence, the solution is correct.