Question

How do you solve the system of equations $4x - 5y = 25$ and $ - 7x + 3y = 8$?

Answer 1

Hint: 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 the other equation and find the value of the remaining one variable.

Complete step-by-step answer:
We are given a pair of linear equations $4x - 5y = 25$ and $ - 7x + 3y = 8$ by applying the method of substitution we will find the value of both variables. We will solve the first equation for $x$:
$ \Rightarrow 4x - 5y = 25$
Adding $5y$ to both sides:
$ \Rightarrow 4x - 5y + 5y = 25 + 5y$
On proper rearrangement we will get:
$ \Rightarrow 4x = 25 + 5y$
Dividing both sides by $4$
$ \Rightarrow x = \dfrac{{25}}{4} + \dfrac{5}{4}y$
Now we will substitute the value of $x$ in the second equation and solve for $y$:
$ \Rightarrow - 7\left( {\dfrac{{25}}{4} + \dfrac{5}{4}y} \right) + 3y = 8$
$ \Rightarrow - \dfrac{{175}}{4} - \dfrac{{35}}{4}y + 3y = 8$
Adding$\dfrac{{175}}{4}$both the sides we will get:
$ \Rightarrow - \dfrac{{175}}{4} - \dfrac{{35y + 12y}}{4} + \dfrac{{175}}{4} = 8 + \dfrac{{175}}{4}$
$ \Rightarrow - \dfrac{{23y}}{4} = \dfrac{{32 + 175}}{4}$
$ \Rightarrow - \dfrac{{23y}}{4} = \dfrac{{207}}{4}$
Multiply both sides by$ - \dfrac{4}{{23}}$:
$ \Rightarrow - \dfrac{{23y}}{4} \times \dfrac{{ - 4}}{{23}} = \dfrac{{207}}{4} \times \dfrac{{ - 4}}{{23}}$
$ \Rightarrow y = - 9$
Substitute $ - 9$ for y in the solution to the expression $x = \dfrac{{25}}{4} + \dfrac{5}{4}y$ and calculate the value of $x$:
$ \Rightarrow x = \dfrac{{25}}{4} + \dfrac{5}{4}\left( { - 9} \right)$
On further solving we will get:
$ \Rightarrow x = \dfrac{{25}}{4} - \dfrac{{45}}{4}$
$ \Rightarrow x = \dfrac{{25 - 45}}{4}$
$ \Rightarrow x = \dfrac{{ - 20}}{4}$
$ \Rightarrow x = - 5$
So the solution is $x = - 5;y = - 9$

Final answer: Hence the solution is $x = - 5;y = - 9$

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.
$4x - 5y = 25$…(1)
$ - 7x + 3y = 8$….(2)
Multiply (1) equation by $3$ and (2) equation by $5$
$12x - 15y = 75$…(3)
$ - 35x + 15y = 40$…(4)
Add equation (3) and (4)
\[
  {\text{ }}{{12x}} - 15y = 75 \\
  \underline {( + ){{ - 35x}} + 15y = 40} \\
  {\text{ }} - 23x = 115 \\
 \]
Now dividing the both sides by$ - 23$
$x = - 5$
Substitute $x = - 5$ in equation (1) we get the value of $y$
$ \Rightarrow 4\left( { - 5} \right) - 5y = 25$
$ \Rightarrow - 5y = 25 + 20$
$ \Rightarrow y = \dfrac{{45}}{{ - 5}}$
$ \Rightarrow y = - 9$
Here also we will get the same solution i.e. $x = - 5;y = - 9$