Question

How do you solve the following the linear equation:
$2x + 3y = 13$, $y - 3 = - 2x$

Answer 1

Hint: The linear equations are the equation which has one order. If the total number of unknown variables is equal to two linear equations, we shall solve them by the elimination method. The elimination method is the method that will eliminate the unknown values and substitute one value to find another unknown value.

Complete step-by-step answer:
Given,
The two linear equations which are given to be solved are
$2x + 3y = 13$
$y - 3 = - 2x$
Let the equation $2x + 3y = 13$by the first equation,
$2x + 3y = 13......(1)$
Let the equation $y - 3 = - 2x$be the second equation,
$y - 3 = - 2x......(2)$
We need to multiply the equation $2$with $ - 1$, we get
$ - 1 \times (y - 3) = - 1 \times - 2x......(2)$
Multiplying we get
$ - y + 3 = 2x.......(3)$
Let the equation $ - y + 3 = 2x$be the third equation.
If we add both the equation, The addition of equation is the equation of left side of both equations are added and the equation of right side of both equations, we get
$1 + 3 \Rightarrow 2x + 3y - y + 3 = 13 + 2x$
Adding respective like terms on the left side of the equation,
$2x + 2y - 3 = 13 + 2x$
By bringing the like terms on the left and right side of the equation,
$2x + 2y - 3 - 2x = 13$
As we add the like terms, we get
$2y - 3 = 13$
The constant values are taken on the right side, we get
$2y = 13 + 3$
Adding the constant values on the right side, we get
$2y = 16$
Bringing the value in the left side to the right side, that is from multiplication to division, we get
\[y = \dfrac{{16}}{2}\]
Dividing the denominator and the numerator, we get
$y = 8$
Substituting $y = 8$in the equation $1$, we get
$2x + 3 \times 8 = 13$
According to the BODMAS rule, first, we should multiply the terms, we get
$2x + 24 = 13$
By bringing the values on the left side of the equation to the right side of the equation,
$2x = 13 - 24$
By subtracting the term,
$2x = - 11$
Bringing the value in the left side to the right side, that is from multiplication to division, we get
$x = - \dfrac{{11}}{2}$
By solving the linear equations, we get
$x = - \dfrac{{11}}{2}$and $y = 8$

Note: Here, we solved the given linear equations by an elimination method. Also, we can apply the substitution method to solve the equations. To solve the substitution method, we need to find the value of any one of the unknown variables. Then, we shall substitute the resultant value in any one of the equations and we need to keep solving to obtain the required equation.