Question

How do you solve \[5x + y = 9\] and \[10x - 7y = - 18\] ?

Answer 1

Hint: The given is a pair of equations with two variables. So we have to solve it to get the value of x and y. For that we will make one of the coefficients of the variable the same. Then we will perform either addition or subtraction such that one of the variables get eliminated. Then we can easily find the value of the remaining variable. Then we will use the given equations to find the value of the eliminated variable.

Complete step-by-step answer:
Given that,
 \[5x + y = 9\] ……eq1
 \[10x - 7y = - 18\] …..eq2
Here we can observe that the sign of y variables coefficients are opposite. This will help in cancellation of the variable easily. Only we need to make the coefficients the same. So we will multiply eq2 by 7.
 \[7\left( {5x + y = 9} \right)\]
On multiplying we get,
 \[35x + 7y = 63\] …..eq3
Now adding eq3 and eq2 such that variable on one side and constants on the other side of the equals to,
 \[35x + 7y + 10x - 7y = 63 + \left( { - 18} \right)\]
Cancelling the y term,
 \[35x + 10x = 63 - 18\]
On performing the operations,
 \[45x = 45\]
To find the value of x,
 \[x = \dfrac{{45}}{{45}}\]
On dividing we get,
 \[x = 1\]
This is the value of x. Now using any of the three equations above we can find the value of y.
So putting the value of x in eq1 we get,
 \[5\left( 1 \right) + y = 9\]
 \[5 + y = 9\]
Transposing constants on one side,
 \[
  y = 9 - 5 \\
  y = 4 \;
\]
This is the value of y.
Thus the solution of the pair of equations is \[x = 1\& y = 4\] .

So, the correct answer is “ \[x = 1\& y = 4\] .”.

Note: Note that solving the equation is nothing but finding a pair of x and y (in this case) that satisfies the equation above. Here we used a method of elimination because the signs of the other variable terms are opposite. So instead of substitution we used elimination method. Also note that the number of variables requires the same number of equations to be solved.