Question

How do you solve $2X - 3Y = 21$ and $5X + 4Y = - 5$?

Answer 1

Hint: Choose either of the two equations, say (i), and find the value of one variable, say $Y$, in terms of the other, i.e., $X$. Then, substitute the value of $Y$, obtained in the other equation, i.e., (ii) to get an equation in $X$ and solve the equation to get the value of $X$. Then, substitute the value of $x$ obtained in the expression for $Y$ in terms of $X$ obtained to get the value of $Y$. The values of $X$ and $Y$ obtained constitute the solution of the given system of two linear equations.

Formula used:
Method of Elimination by Substitution:
In this method, we express one of the variables in terms of the other variable from either of the two equations and then this expression is put in the other equation to obtain an equation in one variable.

Complete step by step solution:
The given system of equations is
$2X - 3Y = 21$…(i)
$5X + 4Y = - 5$…(ii)
Choose either of the two equations, say (i), and find the value of one variable, say $Y$, in terms of the other, i.e., $X$.
$ - 3Y = 21 - 2X$
$ \Rightarrow Y = - 7 + \dfrac{2}{3}X$
Substitute the value of $Y$, obtained in the other equation, i.e., (ii) to get an equation in $X$.
$5X + 4\left( { - 7 + \dfrac{2}{3}X} \right) = - 5$
Solve the equation to get the value of $X$.
$5X - 28 + \dfrac{8}{3}X = - 5$
Add variables and constant terms together.
$ \Rightarrow \dfrac{{15X + 8X}}{3} = - 5 + 28$
$ \Rightarrow \dfrac{{23}}{3}X = 23$
$ \Rightarrow X = 3$
Substitute the value of $X$ obtained in the expression for $Y$ in terms of $X$ obtained to get the value of $Y$.
$Y = - 7 + \dfrac{2}{3} \times 3$
$ \Rightarrow Y = - 5$
The values of $X$ and $Y$ obtained constitute the solution of the given system of two linear equations.
Hence, the solution of the given system of equations is $X = 3$, $Y = - 5$.

Note:
We can also find the solution of a given system by Method of Elimination by equating the coefficients.
Method of Elimination by equating the coefficients:
In this method, we eliminate one of the two variables to obtain an equation in one variable which can be easily solved. Putting the value of this variable in any one of the given equations, the value of another variable can be obtained.
Step by step solution:
The given system of equations is
$2X - 3Y = 21$…(i)
$5X + 4Y = - 5$…(ii)
We can eliminate the $Y$ variable by multiplying the first equation by $4$, and second equation by $3$ and then adding the two equations.
Multiply both sides of equation (i) by $4$, we get
$8X - 12Y = 84$…(iii)
Multiply both sides of equation (ii) by $3$, we get
$15X + 12Y = - 15$…(iv)
Now, add equations (iii) and (iv), we get
$ \Rightarrow 23X = 69$
Now, multiply both sides of the equation by $3$.
$ \Rightarrow X = 3$
Now, substitute the value of $X$ in equation (ii) and find the value of $Y$.
$ \Rightarrow 15 + 4Y = - 5$
$ \Rightarrow 4Y = - 20$
$ \Rightarrow Y = - 5$
Final solution: Hence, the solution of the given system of equations is $X = 3$, $Y = - 5$.