Question

How do you solve $3x + 4y = 7$and$5x - 2y = 16$?

Answer 1

Hint: According to the question we have to solve the given linear equation with two variables which are x and y are $3x + 4y = 7$ and $5x - 2y = 16$. So, to solve the given linear expressions or we can say to determine the values of x and y first of all we have to make any one of the given variable equal to the other which can be done by multiplying 2 in the given linear expression $5x - 2y = 16$ so that we can make the variable y similar for both of the expression to eliminate.
Now, we have to add the both of the expressions one of which is obtained after the multiplication with 2 to obtain the value of the variable x.
Now, we have to substitute the value of the variable x in any one of the expressions so that we can also obtain the value of the variable y.

Complete step by step answer:
Step 1: First of all we have to make any one of the given variables equal to the other which can be done by multiplying 2 in the given linear expression $5x - 2y = 16$ so that we can make the variable y similar for both of the expressions to eliminate. Hence, on multiplying it with 2,
$
   \Rightarrow 2(5x - 2y) = 2 \times 16 \\
   \Rightarrow 10x - 4y = 32.............(1) \\
 $
And, the other linear expression is:
$ \Rightarrow 3x + 4y = 7...............(2)$
Step 2: Now, we have to add the both of the expressions one of which is obtained after the multiplication with 2 to obtain the value of the variable x. Hence,
$
   \Rightarrow 10x - 4y + 3x + 4y = 32 + 7 \\
   \Rightarrow 13x = 39 \\
 $
On applying cross-multiplication in the expression as obtained just above,
$
   \Rightarrow x = \dfrac{{39}}{{13}} \\
   \Rightarrow x = 3 \\
 $
Step 3: Now, we have to substitute the value of the variable x in any one of the expressions so that we can also obtain the value of the variable y. Hence, on substituting the obtained value of x in the expression $3x + 4y = 7$,
$
   \Rightarrow 3(3) + 4y = 7 \\
   \Rightarrow 4y = 7 - 9 \\
   \Rightarrow 4y = - 2 \\
 $
On applying cross-multiplication in the expression as obtained just above,
$
   \Rightarrow y = \dfrac{{ - 2}}{4} \\
   \Rightarrow y = - \dfrac{1}{2} \\
 $

Hence, with the help of elimination and substitution we have determined the values of the given expressions which are $x = 3$ and $y = - \dfrac{1}{2}$.

Note: Elimination method is the method in which we have to eliminate the given variable in the expression which can be x and y or some other variable by adding or subtracting both of the expressions.
In the substitution method it is necessary that we have to substitute the value of the variable obtained in the other expression to determine the value of the other variable.