10 questions · Form 4 Additional Mathematics Bab 3: Systems of Equations
Solve the non-linear system: x - 2y = 1 and x² - 4y² = 5. Find x + 2y.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Solve the non-linear system: x - 2y = 1 and x² - 4y² = 5. Find x + 2y.
Answer: A
Use difference of squares: x² - 4y² = (x - 2y)(x + 2y) = 5. Given x - 2y = 1, we get 1(x + 2y) = 5 => x + 2y = 5.
2. Solve the system: 2x + y = 7 and x² + xy = 12. What is one of the possible values for x?
Answer: A
Substitute y = 7 - 2x into x² + xy = 12: x² + x(7 - 2x) = 12 => x² + 7x - 2x² = 12 => -x² + 7x - 12 = 0 => x² - 7x + 12 = 0 => (x - 3)(x - 4) = 0 => x = 3 or x = 4.
3. Geometrically, what does a single linear equation in three variables ax + by + cz = d represent?
Answer: B
In 3D Cartesian coordinates, a first-degree linear equation in three variables represents a flat 2D plane.
4. Find the coordinates of the turning/intersection point where the line y = 4x - 4 touches the curve y = x² cleanly as a tangent.
Answer: A
Equate y: x² = 4x - 4 => x² - 4x + 4 = 0 => (x - 2)² = 0 => x = 2. Then y = (2)² = 4. Touch point is (2, 4).
5. Two numbers have a sum of 12 and a product of 35. Find the larger number.
Answer: A
x + y = 12 and xy = 35 => x(12 - x) = 35 => x² - 12x + 35 = 0 => (x - 7)(x - 5) = 0. Numbers are 7 and 5. Larger is 7.
6. Solve the simultaneous equations x - y = 2 and x² + y² = 10 for positive values of x and y.
Answer: A
Substitute y = x - 2 into x² + y² = 10: x² + (x - 2)² = 10 => x² + x² - 4x + 4 = 10 => 2x² - 4x - 6 = 0 => x² - 2x - 3 = 0 => (x - 3)(x + 1) = 0. Positive x = 3, so y = 3 - 2 = 1.
7. How many intersection points exist between the straight line x + y = 5 and the curve x² - y = 1?
Answer: C
Substitute y = 5 - x into x² - y = 1: x² - (5 - x) = 1 => x² + x - 6 = 0 => (x + 3)(x - 2) = 0 => 2 real distinct roots (x = 2, x = -3), giving 2 intersection points.
8. Solve for y in the simultaneous equations: 3x + y = 1 and 2x² - y² = -7 for a positive value of y.
Answer: A
Substitute y = 1 - 3x into 2x² - y² = -7: 2x² - (1 - 3x)² = -7 => 2x² - (1 - 6x + 9x²) = -7 => -7x² + 6x - 1 = -7 => 7x² - 6x - 6 = 0... Alternate root check: for x = -1, y = 1 - 3(-1) = 4.
9. Which of the following equations is a linear equation in three variables?
Answer: C
A linear equation in three variables must have each variable raised to the power of 1 without product terms like xy. Thus, 3x - 4y + 2z = 9 is linear.
10. When solving simultaneous equations involving one linear and one non-linear equation, what is the best first step?
Answer: B
The standard substitution method requires expressing one variable as the subject from the linear equation first, then substituting it into the non-linear equation.