Question

How do you solve \[4x-y=-8\] and \[ x+3y=-17\]?

Answer 1

Hint: In the given question, we have been asked to solve a system of equation i.e. \[4x-y=-8\] and \[ x+3y=-17\]. In order to solve the equations we will use elimination methods. We need to either add both the equations or subtract both the equations to get the equation in one variable. Then we solve the equation in one variable in a way we solve the general linear equation.

Complete step by step solution:
We have given that
\[\Rightarrow 4x-y=-8\]------ (1)
\[\Rightarrow \ x+3y=-17\]----- (2)
Multiply the equation (2) by 4, we get
\[\Rightarrow 4\ x+12y=-68\]----- (3)
Subtracting equation (3) from equation (1), we get
\[\Rightarrow 4x-y-\left( 4\ x+12y \right)=-8-\left( -68 \right)\]
Simplifying the above equation, we get
\[\Rightarrow 4x-y-4x-12y=-8+68\]
Combining the like terms, we get
\[\Rightarrow -13y=-8+68\]
Simplifying the numbers in the above equation, we get
\[\Rightarrow -13y=60\]
Dividing both the sides of the equation by -3, we get
\[\Rightarrow y=-\dfrac{60}{13}\]
Substitute the value of \[y=-\dfrac{60}{13}\] in equation (1), we get
\[\Rightarrow 4x-y=-8\]
\[\Rightarrow 4x-\left( -\dfrac{60}{13} \right)=-8\]
\[\Rightarrow 4x+\dfrac{60}{13}=-8\]
Subtracting \[\dfrac{60}{13}\] from both the sides of the equation, we get
\[\Rightarrow 4x+\dfrac{60}{13}-\dfrac{60}{13}=-8-\dfrac{60}{13}\]
Simplifying the numbers in the above equation, we get
\[\Rightarrow 4x=-8-\dfrac{60}{13}\]
Simplifying the right-hand side of the equation by taking LCM, we get
LCM of 13 and 1 is 13.
\[\Rightarrow 4x=-\dfrac{104}{13}-\dfrac{60}{13}\]
Simplifying the numbers in the above equation, we get
\[\Rightarrow 4x=-\dfrac{164}{13}\]
Multiply both the sides of the equation by 13, we get
\[\Rightarrow 52x=-164\]
Dividing both the side of the equation by -52, we get
\[\Rightarrow x=-\dfrac{164}{52}\]
Converting the fraction into simplest form, we get
\[\Rightarrow x=-\dfrac{41}{13}\]
Therefore, we get
\[\Rightarrow \left( x,y \right)=\left( -\dfrac{41}{13},-\dfrac{60}{13} \right)\]
Therefore, the possible value of ‘x’ and ‘y’ is\[\left( x,y \right)=\left( -\dfrac{41}{13},-\dfrac{60}{13} \right)\].
It is the required solution.

Note: In an elimination method for solving a system of equations we will either add both the equations or subtract both the equations to get the equation in one variable. To eliminate the variable, you will add the two given equations if the coefficients of one variable are opposites and you will subtract the two equations if the coefficient of one variable is exactly the same. If both the situations do not satisfy then we will need to multiply one of the equations by a number that will lead to the same leading coefficient.