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How do you solve the linear system using substitution $x = 16 - 4y$ and $3x + 4y = 8$?

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Last updated date: 14th Jun 2024
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Answer
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Hint: Take out all the like terms to one side and all the alike terms to the other side. Take out all the common terms. Then substitute the value of the expression in the other equation. Reduce the terms on the both sides until they cannot be reduced any further if possible. Then finally evaluate the value of the unknown variable.

Complete step-by-step solution:
First we will start off by taking all the like terms to one side in the first equation.
$x = 16 - 4y$
Now we will substitute the first equation in the second equation.
$
\Rightarrow 3x + 4y = 8 \\
\Rightarrow 3(16 - 4y) + 4y = 8 \\
 $
Now we will open the brackets and reduce the terms on both the sides.
$
\Rightarrow 3(16 - 4y) + 4y = 8 \\
\Rightarrow 48 - 12y + 4y = 8 \\
\Rightarrow -8y = - 40 \\
 $
Now we simplify our final answer that is evaluate the value of the variable $y$.
$
\Rightarrow - 8y = - 40 \\
\Rightarrow 8y = 40 \\
\Rightarrow y = \dfrac{{40}}{8} \\
 $
Hence, the value of $y$ is $\dfrac{{40}}{8}$.

Additional Information: To cross multiply terms, you will multiply the numerator in the first fraction times the denominator in the second fraction, then you write that number down. Then you multiply the numerator of the second fraction times the number in the denominator of your first fraction, and then you write that number down. By Cross multiplication of fractions, we get to know if two fractions are equal or which one is greater. This is especially useful when you are working with larger fractions that you are not sure how to reduce. Cross multiplication also helps us to solve for unknown variables in fractions.

Note: While taking terms from one side to another, make sure you are changing their respective signs as well. While opening any brackets, always multiply the signs present outside the brackets along with the terms. Reduce the terms using the factorisation method.