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Solve the following system of equations and show all work
y = 2x^2
у= -3× - 1

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Final answer:

The system of equations y = 2x^2 and y = -3x - 1 is solved using the substitution method.

We set the expressions equal to each other and solve the resulting quadratic equation to find the x-values.

The solutions to the system are (-1/2, 1/2) and (-1, 2).

Step-by-step explanation:

To solve the system of equations y = 2x^2 and y = -3x - 1, we will use the substitution method since both equations are already solved for y.

This implies setting the two expressions for y equal to each other and solving for x.

  • Set 2x^2 equal to -3x - 1:
  • 2x^2 = -3x - 1
  • Rearrange the equation to form a quadratic equation:
  • 2x^2 + 3x + 1 = 0
  • Factor the quadratic equation (if possible) or use the quadratic formula to find the x-values.
  • In this case, the equation factors to:
  • (2x + 1)(x + 1) = 0
  • Set each factor equal to zero and solve for x:
  • x = -1/2 or x = -1
  • Substitute the x-values back into one of the original equations to find the corresponding y-values:
  • For x = -1/2:
  • y = 2(-1/2)^2 = 1/2
  • For x = -1:
  • y = 2(-1)^2 = 2
  • Thus, we have two solutions for the system:
  • (-1/2, 1/2) and (-1, 2)
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