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# How to solve for $y$ in $5x - y = 33$ and $7x + y = 51$?

Last updated date: 13th Jun 2024
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Hint:
Here, we will use the method of elimination to find the solution of the given equations. We will first add the given equations and simplify it to find the value of variable $x$. Then we will substitute this value in one of the equations to get the value of $y$.

Complete step by step solution:
The given linear equations are:
$5x - y = 33$ ……………………………………………………$\left( 1 \right)$
$7x + y = 51$ …………..………………………………………$\left( 2 \right)$
Now, we will add equations $\left( 1 \right)$ and $\left( 2 \right)$. Therefore, we get
$5x - y + 7x + y = 33 + 51$
Adding and subtracting the like terms, we get
$\Rightarrow 12x = 84$
Dividing both sides by 12, we get
$\Rightarrow x = \dfrac{{84}}{{12}}$
$\Rightarrow x = 7$
Now, by substituting $x = 7$ in equation $\left( 1 \right)$, we get
$5\left( 7 \right) - y = 33$
Multiplying the terms, we get
$\Rightarrow 35 - y = 33$
Now, by rewriting the equation, we get
$\Rightarrow y = 35 - 33$
Subtracting the terms, we get
$\Rightarrow y = 2$

Therefore, the solution for the $5x - y = 33$ and $7x + y = 51$ is $x = 7$ and $y = 2$