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How to solve for \[y\] in \[5x - y = 33\] and \[7x + y = 51\]?

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Last updated date: 05th Mar 2024
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IVSAT 2024
Answer
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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 \[y = 2\]

Additional Information:
The solution set for the linear equations of two variables and with only one equation can be obtained only by the method of substitution. But we know that the linear equation of two variables can be solved by elimination method, cross multiplication method and substitution method. We are using the method of elimination where one variable is eliminated either by adding or subtracting the equations.

Note:
We know that an equation is defined as a mathematical statement with an equality sign between the two algebraic expressions. Linear equations are a combination of constants and variables. Linear equation is defined as an equation with the highest degree as 1. We know that the solution set is a set of values which satisfies the relation between the two mathematical expressions.
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