To eliminate the y-terms and solve for x in the fewest steps, by which constants should the equations be multiplied by before adding the equations together?

Question

First Equation: 5x − 4y = 28

Second equation: 3x - 9y = 30

The first equation should be multiplied by 3 and the second equation by 5.
The first equation should be multiplied by 3 and the second equation by −5.
The first equation should be multiplied by 9 and the second equation by 4.
The first equation should be multiplied by 9 and the second equation by −4.

Answer 1

To eliminate the y-terms efficiently, we can manipulate the coefficients of the \(y\) terms in both equations.

The first equation is: \[ 5x - 4y = 28 \] The second equation is: \[ 3x - 9y = 30 \]

The coefficient of \(y\) in the first equation is \(-4\) and in the second equation, it is \(-9\).

To eliminate \(y\) by multiplying the first equation by \(9\) and the second equation by \(4\), we can set the equations as follows:

1. Multiply the first equation by \(9\): \[ 9(5x - 4y) = 9(28) \] This gives: \[ 45x - 36y = 252 \]

2. Multiply the second equation by \(4\): \[ 4(3x - 9y) = 4(30) \] This gives: \[ 12x - 36y = 120 \]

Now, if we add these two modified equations, the \(y\) terms will cancel out: \[ (45x - 36y) + (12x - 36y) = 252 + 120 \implies 57x - 72y = 372 \text{ (We notice term cancellation)} \]

The correct way that involves eliminating \(y\) effectively is:

The first equation should be multiplied by \(9\) and the second equation by \(4\).

Thus, the correct option is: The first equation should be multiplied by 9 and the second equation by 4.