Math Problem Statement

Solve the nonlinear system of equations for real solutions.

left brace Start 2 By 1 Matrix 1st Row 1st Column y equals x squared minus 7 2nd Row 1st Column y equals x squared minus 7 x EndMatrix

y=x2−7

y=x2−7x

Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The​ solution(s) is/are

enter your response here.

​(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed. Type exact answers for each​ coordinate, using radicals as​ needed.)

B.

There is no solution.

Solution

To solve the given nonlinear system of equations:

  1. y=x2−7y = x^2 - 7
  2. y=x2−7xy = x^2 - 7x

we will follow these steps:

Step 1: Set the equations equal to each other.

Since y=x2−7y = x^2 - 7 and y=x2−7xy = x^2 - 7x, we can equate the right-hand sides:

x2−7=x2−7xx^2 - 7 = x^2 - 7x

Step 2: Simplify the equation.

Subtract x2x^2 from both sides:

−7=−7x-7 = -7x

Divide through by −7-7 (assuming x≠0x \neq 0):

x=1x = 1

Step 3: Solve for yy using x=1x = 1.

Substitute x=1x = 1 into either equation. Using y=x2−7y = x^2 - 7:

y=(1)2−7=1−7=−6y = (1)^2 - 7 = 1 - 7 = -6

Thus, one solution is:

(1,−6)(1, -6)

Step 4: Check for other solutions.

We must verify if there are other solutions. Return to the two original equations:

  1. y=x2−7y = x^2 - 7
  2. y=x2−7xy = x^2 - 7x

Substitute y=x2−7y = x^2 - 7 into the second equation:

x2−7=x2−7xx^2 - 7 = x^2 - 7x

As shown earlier, this simplifies to x=1x = 1. Therefore, no other solutions exist.

Final Answer:

The solution is:

A. The solution is (1,−6)(1, -6).

Verification:

  1. For x=1x = 1:
    • y=12−7=−6y = 1^2 - 7 = -6
    • y=12−7(1)=−6y = 1^2 - 7(1) = -6

Both equations are satisfied. There are no additional solutions.

Questions for Deeper Understanding:

  1. How can you graphically interpret the intersection of the two curves?
  2. What happens if we change the constants in the equations slightly?
  3. How would this problem differ if the second equation were y=x2+7xy = x^2 + 7x?
  4. What is the significance of the solution (1,−6)(1, -6) in the context of these equations?
  5. How do nonlinear systems differ from linear systems in terms of solving methods?

Tip:

Always check for extraneous solutions when solving systems involving nonlinear equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Nonlinear Systems of Equations

Formulas

Equating two functions to find common solutions

Theorems

Properties of quadratic functions

Suitable Grade Level

Grades 10-12

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