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$2x+5y=10$ and $3y+4x=12$ . find $x$ and $y$

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Composition of linear functions: find a value

Full solution

Q.

2x+5y=10

and

3y+4x=12

. find

x

and

y

Write Equations: Write down the system of equations. $\newline$ We have the following system of equations: $\newline$ $1$ ) $2x + 5y = 10$ $\newline$ $2$ ) $4x + 3y = 12$
Eliminate y: Multiply the first equation by $2$ and the second equation by $-5$ to eliminate $y$ . $\newline$ Multiplying the first equation by $2$ : $\newline$ $(2x + 5y) \times 2 = 10 \times 2$ $\newline$ $4x + 10y = 20$ $\newline$ Multiplying the second equation by $-5$ : $\newline$ $(4x + 3y) \times -5 = 12 \times -5$ $\newline$ $-20x - 15y = -60$
Add Equations: Add the new equations together to eliminate $y$ . $\newline$ Adding the equations $4x + 10y = 20$ and $-20x - 15y = -60$ : $\newline$ $(4x + 10y) + (-20x - 15y) = 20 + (-60)$ $\newline$ $4x - 20x + 10y - 15y = 20 - 60$ $\newline$ $-16x - 5y = -40$
Correct Mistake: Since we made a mistake in the previous step by adding $y$ terms instead of canceling them out, we need to correct this. $\newline$ Let's add the equations again, but this time we will focus on canceling the $y$ terms: $\newline$ $(4x + 10y) + (-20x - 15y) = 20 + (-60)$ $\newline$ $4x - 20x + 10y - 15y = 20 - 60$ $\newline$ $-16x - 5y = -40$ $\newline$ This is incorrect because we should have: $\newline$ $4x - 20x = -16x$ $\newline$ $10y - 15y = -5y$ $\newline$ So the correct equation after adding should be: $\newline$ $-16x - 5y = -40$ $\newline$ However, this does not help us eliminate $y$ . We need to go back and correctly eliminate $y$ by adding the right multiples of the original equations.

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