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Use the substitution method to solve the system 2x + 3y = 4, - 4х - 6y = -8.

(1 point)
• infinite solutions
• no solution
• (2 - 1)
O (-1,2)

1 answer

To solve the system of equations using the substitution method, let's first rewrite the equations clearly:

  1. \( 2x + 3y = 4 \)
  2. \( -4x - 6y = -8 \)

Let's simplify the second equation, \( -4x - 6y = -8 \). Notice that if we multiply the entire first equation by 2, we get:

\[ 2(2x + 3y) = 2(4) \implies 4x + 6y = 8 \tag{3} \]

Now, let's compare equations 2 and 3.

  1. \( -4x - 6y = -8 \)
  2. \( 4x + 6y = 8 \)

If we multiply equation 2 by -1:

\[ 4x + 6y = 8 \]

This is the same as equation 3. So we see that equation 2 is a multiple of equation 1. This means that the two equations represent the same line.

Because they represent the same line, there are infinitely many solutions to this system of equations.

Thus, the correct answer is infinite solutions.