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SAT Math: Nonlinear Equations (Drill 1)

Drill 1 · Math · Nonlinear Equations

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About This Drill

SAT Math: Nonlinear Equations (Drill 1) is a Math practice drill covering Nonlinear Equations. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

SAT nonlinear equation questions involve systems combining a linear and a quadratic equation, radical equations, rational equations, and higher-degree polynomials. Remember to check for extraneous solutions introduced by squaring or clearing denominators.

Questions & Explanations

Question 1. y = x + 2 and y = x². The system of equations above has two solutions. What is the sum of the y-coordinates of those solutions?

  • A) 3
  • B) 4
  • C) 1
  • D) 5 ✓

Explanation: Substitute y = x + 2 into y = x²: x + 2 = x². Rearrange: x² − x − 2 = 0. Factor: (x − 2)(x + 1) = 0. So x = 2 (giving y = 4) and x = −1 (giving y = 1). The sum of the y-coordinates is 4 + 1 = 5.

Question 2. If √3x + 1 = 5, what is the value of x?

  • A) 4
  • B) 8 ✓
  • C) 12
  • D) 24

Explanation: Square both sides: 3x + 1 = 25. Subtract 1: 3x = 24. Divide by 3: x = 8. Check: √3(8) + 1 = √25 = 5. ✓

Question 3. What is the solution to √x + 3 = x − 3?

  • A) x = 1 only
  • B) x = 1 and x = 6
  • C) x = 6 only ✓
  • D) No solution

Explanation: Square both sides: x + 3 = (x − 3)² = x² − 6x + 9. Rearrange: x² − 7x + 6 = 0. Factor: (x − 1)(x − 6) = 0. Check x = 1: √4 = 2 but x − 3 = −2. Since 2 ≠ −2, x = 1 is extraneous. Check x = 6: √9 = 3 and x − 3 = 3. ✓ Only x = 6 is valid.

Question 4. If 6/(x − 1) = 3, what is the value of x?

  • A) 3 ✓
  • B) 2
  • C) 5
  • D) 7

Explanation: Multiply both sides by (x − 1): 6 = 3(x − 1). Distribute: 6 = 3x − 3. Add 3: 9 = 3x. Divide: x = 3. Check: 6/(3 − 1) = 6/2 = 3. ✓

Question 5. y = x² + 1 and y = 3. How many solutions does the system of equations above have?

  • A) Zero
  • B) Two ✓
  • C) One
  • D) Infinitely many

Explanation: Substitute: x² + 1 = 3. So x² = 2, giving x = √2 and x = −√2. The horizontal line y = 3 intersects the parabola y = x² + 1 at two points.