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Difficulty: EasyNonlinear Systems of Equations

A system of two equations is shown below.

y=x2−4y = x^2 - 4
y=x−2y = x - 2

Which of the following ordered pairs (x,y)(x, y) is a solution to the system?

  1. (2,0)(2, 0)Answer
  2. B
    (0,−2)(0, -2)
  3. C
    (−2,0)(-2, 0)
  4. D
    (1,−3)(1, -3)

Answer

The ordered pair (2,0)(2, 0) is the correct solution to the system.
The ordered pair (2,0)(2, 0) is the correct solution because substituting these values into both equations in the system results in true statements. For the quadratic equation, 0=22−40 = 2^2 - 4 simplifies to 0=00 = 0. For the linear equation, 0=2−20 = 2 - 2 simplifies to 0=00 = 0.

Step-by-Step Solution

1
Substitute the xx-value and yy-value from the candidate solution (2,0)(2, 0) into the first equation, y=x2−4y = x^2 - 4.
0=22−4  ⟹  0=4−4  ⟹  0=00 = 2^2 - 4 \implies 0 = 4 - 4 \implies 0 = 0, which is a true statement.
An ordered pair must satisfy the first equation to be a candidate solution for the system.
2
Substitute the xx-value and yy-value from the candidate solution (2,0)(2, 0) into the second equation, y=x−2y = x - 2.
0=2−2  ⟹  0=00 = 2 - 2 \implies 0 = 0, which is also a true statement.
An ordered pair must satisfy all equations in the system simultaneously to be a valid solution.

Key Concept

Verifying a solution to a nonlinear system of equations by substituting the coordinate values into both equations.
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Topics

Nonlinear Systems of Equations

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