Fill in the blank so that the resulting statement is true.

Question

Consider the following system. When solving the system by the addition​ method, we can eliminate y by multiplying the second equation by​ ____ and then adding the equations.
left brace Start 2 By 1 Matrix 1st Row 1st Column 2 x plus 16 y equals 11 2nd Row 1st Column 4 x plus 4 y equals 10 EndMatrix

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Part 1
When solving the system by the addition​ method, we can eliminate y by multiplying the second equation by
negative 4
negative 1
negative 5
negative 2
negative 4
and then adding the equations.

Answer 1

To eliminate \(y\) using the addition method, we need the coefficients of \(y\) in both equations to be opposites.

Let's start by rewriting the given equations:

1. \(2x + 16y = 11\) (Equation 1)
2. \(4x + 4y = 10\) (Equation 2)

Currently, the coefficient of \(y\) in Equation 1 is \(16\) and in Equation 2 it is \(4\).

To eliminate \(y\), we want the coefficients to be equal in magnitude but opposite in sign. This means we can multiply the entirety of Equation 2 by \(-4\):

\[ -4(4x + 4y) = -4(10) \ -16x - 16y = -40 \]

Now we have the equations:

1. \(2x + 16y = 11\)
2. \(-16x - 16y = -40\)

Now, if we add these two equations together, the \(16y\) from Equation 1 and \(-16y\) from Equation 2 will cancel out, allowing us to eliminate \(y\).

So the answer is: negative 4.