To use the elimination method to eliminate one of the variables, we should manipulate the equations so that when we add or subtract them, one variable can be canceled out.
Let's start with the given equations:
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\( 6x + 3y = 14 \)
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\( 3x + 2y = 10 \)
To eliminate \(y\), we can make the coefficients of \(y\) in both equations equal. Currently, in the first equation, the coefficient of \(y\) is 3, and in the second equation, it is 2.
We can multiply the second equation by 1.5 (or \(\frac{3}{2}\)) to make the coefficient of \(y\) in the second equation equal to 3:
\[ 1.5(3x + 2y) = 1.5(10) \]
This gives us:
\[ 4.5x + 3y = 15 \]
Now, we can set this new equation against the first equation:
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\( 6x + 3y = 14 \)
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\( 4.5x + 3y = 15 \)
However, this method is a little more complex than needed. Instead, let’s see if we can simply multiply the second equation by 3 to have \(6x + 3y\) match directly with the first equation. This will eliminate \(x\):
Original equations:
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\( 6x + 3y = 14 \)
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\( 3x + 2y = 10 \)
By multiplying the second equation by 2:
\[ 2(3x + 2y) = 2(10) \]
This results in:
\[ 6x + 4y = 20 \]
Now, we use:
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\( 6x + 3y = 14 \)
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\( 6x + 4y = 20 \)
If we subtract these two equations, \(6x\) cancels out:
Subtract the first from the second:
\[ (6x + 4y) - (6x + 3y) = 20 - 14 \]
This simplifies to:
\[ y = 6 \]
Thus, to eliminate \(y\), Tatum should multiply the second equation by 2.