Question

How do solve the following linear system: $-x+5y=-13,-4x-2y=14$ ?

Answer 1

Hint: For solving the given linear systems, we will use The Elimination Method. In The Elimination Method, we will make both the equation in the form of $ax+by=c$ , where $a$ , $b$ and $c$ are constants. Then, we will have to make coefficients of any one of the both variables that are $x$ and $y$ same. After that we will add or subtract the one equation to the other equation so that one variable might be canceled or eliminated. Since, we got the value of one variable from the above step; we will put that value in any equation in such a way that we can get the value of another variable.

Complete step by step solution:
Since, we have the linear equations as:
\[\Rightarrow -x+5y=-13\] … $\left( i \right)$
And
\[\Rightarrow -4x-2y=14\] … $\left( ii \right)$
Now, we will multiply by $4$ in equation $\left( i \right)$ to make the coefficient of $x$ same as:
\[\Rightarrow \left( -x+5y=-13 \right)\times 4\]
Then, we will open the bracket and will complete the multiplication process to complete the above step as:
\[\Rightarrow -x\times 4+5y\times 4=-13\times 4\]
\[\Rightarrow -4x+20y=-52\] … $\left( iii \right)$
Since, the coefficient of the equation $\left( ii \right)$ and $\left( ii \right)$ are same and the both coefficient have same sign also, we will subtract equation $\left( ii \right)$ from equation $\left( ii \right)$a s:
\[\Rightarrow \left( -4x+20y \right)-\left( -4x-2y \right)=-52-14\]
After opening the above equation, the sign of second bracketed terms will be changed because of a negative sign outside the bracket. Then, we will see that the above equation will be as:
\[\Rightarrow -4x+20y+4x+2y=-52-14\]
Now, we will combine the equal like terms as:
\[\Rightarrow \left( -4x+4x \right)+\left( 20y+2y \right)=\left( -52-14 \right)\]
By adding or subtracting according to the above step, we will find that the $x$ terms are cancel out from the above equation and others will be as:
\[\Rightarrow 22y=-66\]
Now, above equation will be written as:
\[\Rightarrow y=\dfrac{-66}{22}\]
Here, we will have the value of $y$ as:
\[\Rightarrow y=-3\]
For getting the value of $x$, we can use the value of $y$ in any above equation. Let’s put this value in equation $\left( i \right)$ as the equation is:
\[\Rightarrow -x+5y=-13\]
\[\Rightarrow -x+5\left( -3 \right)=-13\]
Here, we will solve the above equation as:
\[\Rightarrow -x-15=-13\]
Since, the signs are opposite so we got the negative sign as a result. Now, we will place equal like terms one side to get the value of $x$ as:
\[\Rightarrow x=-15+13\]
Using subtraction, we will get the value of $x$ :
\[\Rightarrow x=-2\]
Hence, we have \[x=-2\] and \[y=-3\] as resultant.

Note: In the Elimination Method, we need to take care of some steps that are the key point for getting the solution. First of all remember that for getting the same coefficient of variable in both equations, sometimes we might have to multiply in both equations by any number so that we can eliminate or cancel out one variable and sometimes we might have to divide by any number in either one or both equation to cancel out a variable in linear system.