Question

Solve for $x$ and $y$ :
$7\left( y+3 \right)-2\left( x+2 \right)=14$
$4\left( y-2 \right)+3\left( x-3 \right)=2$

Answer 1

Hint: Here we have been given two equations and we have to find the value of two unknown variables in them. Firstly we will simplify the equation by removing the brackets. Then by using the elimination method where we multiply each equation by a number and add them such that one variable is removed we will get the value of one variable. Finally we will put that value in one of the equations and get the value of another variable and our desired answer.

Complete step by step answer:
The two equations are given as follows,
$7\left( y+3 \right)-2\left( x+2 \right)=14$….$\left( 1 \right)$
$4\left( y-2 \right)+3\left( x-3 \right)=2$……$\left( 2 \right)$
On simplifying equation (1) we get,
$\Rightarrow 7y+21-2x-4=14$
$\Rightarrow 7y-2x+17=14$
Taking the constant on right side we get,
$\Rightarrow 7y-2x=14-17$
$\Rightarrow 7y-2x=-3$….$\left( 3 \right)$
Next we will simplify equation (2),
$\Rightarrow 4y-8+3x-9=2$
$\Rightarrow 4y+3x-17=2$
Taking the constant on right side we get,
$\Rightarrow 4y+3x=2+17$
$\Rightarrow 4y+3x=19$…..$\left( 4 \right)$
So we get the two simplified equation as $7y-2x=-3$ and $4y+3x=19$
Now we will use the elimination method to solve the above two equations.
We will make the coefficient of $x$ variable same in both by multiplying equation (3) by $3$ and equation (4) by $2$ as follows,
Multiply equation (3) by $3$ ,
$\Rightarrow \left( 7y-2x=-3 \right)\times 3$
$\Rightarrow 21y-6x=-9$….$\left( 5 \right)$
Multiply equation (4) by $2$
$\Rightarrow \left( 4y+3x=19 \right)\times 2$
$\Rightarrow 8y+6x=38$….$\left( 6 \right)$
On adding equation (5) and (6) we get,
$\begin{align}
  & 21y-6x=-9 \\
 & \underline{+8y+6x=38} \\
 & 29y+0=29 \\
\end{align}$
We get the value as,
$\Rightarrow 29y=29$
$\Rightarrow y=1$
Put $y=1$ in equation (3) and simplify,
$\Rightarrow 7\times 1-2x=-3$
$\Rightarrow 7-2x=-3$
Simplifying further,
$\Rightarrow 7+3=2x$
$\Rightarrow x=\dfrac{10}{2}$
So we get,
$\Rightarrow x=5$
Hence on $7\left( y+3 \right)-2\left( x+2 \right)=14$ and $4\left( y-2 \right)+3\left( x-3 \right)=2$ we get the answer as $x=5$ and $y=1$ .

Note:
 If we are given two linear equations then we can solve them in many ways like using the substitution method and matrix method alternatively.
Elimination method is used to either add or subtract the two equations such that we get an equation in one variable. As the coefficient of the variable $x$ were of opposite sign in the two equations we have added them after making the coefficient value same but if both the variable i.e. $x$ and $y$ sign is same in the two equation in that case we subtract the two equations.