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Michiko claims that to solve the system of linear equations 15 x + 8 y = 1 and 21 x + 4 y = negative 13, instead of multiplying the second equation by –2 and then adding the equations, she could multiply the second equation by 2 and then subtract the second equation from the first equation. Which statement is correct?

Michiko is right for this system of linear equations, but her method will not work for other systems of linear equations.
Michiko is right for this system of linear equations, and her method will also work for other systems of linear equations.
Michiko is wrong because her method will give the opposite of the correct value for x for any system of linear equations.
Michiko is wrong because her method will give the opposite of the correct value for y for any system of linear equations.

1 answer

To analyze Michiko's approach, let's first clarify the two methods she is considering:

  1. Original Proposed Method: Multiply the second equation (21x + 4y = -13) by -2, and then add it to the first equation (15x + 8y = 1).

  2. Alternative Method: Multiply the second equation by 2, and then subtract the second equation from the first.

Method 1: Add after multiplying by -2

  1. First equation:
    \( 15x + 8y = 1 \)
  2. Second equation multiplied by -2:
    \( -2(21x + 4y) = -2(-13) \)
    \( -42x - 8y = 26 \)

Now, adding these two equations:
\[ (15x + 8y) + (-42x - 8y) = 1 + 26 \] \[ -27x = 27 \implies x = -1 \]

Method 2: Subtract after multiplying by 2

  1. First equation:
    \( 15x + 8y = 1 \)
  2. Second equation multiplied by 2:
    \( 2(21x + 4y) = 2(-13) \)
    \( 42x + 8y = -26 \)

Now, subtracting the second equation from the first:
\[ (15x + 8y) - (42x + 8y) = 1 - (-26) \] \[ 15x + 8y - 42x - 8y = 1 + 26 \] \[ -27x = 27 \implies x = -1 \]

Conclusion:

Both methods yield the same result for \(x\). Thus, Michiko's alternative method does work for this specific system of equations as well as possibly for other systems.

For her statement,

  • Correct Statement: Michiko is right for this system of linear equations, and her method will also work for other systems of linear equations.

However, it's important to note that while this method works for this case, whether it works for all other systems depends on the specific parameters of those systems.

Therefore, the best statement to choose would be: Michiko is right for this system of linear equations, and her method will also work for other systems of linear equations.