10 questions · Form 4 Additional Mathematics Bab 3: Systems of Equations
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Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. If elimination yields the identity 0 = 0 when solving a system of three linear equations, what can be concluded?
Answer: B
A tautology such as 0 = 0 signifies dependent equations, meaning the planes intersect along a line or coincide, giving infinitely many solutions.
2. 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.
3. If three planes in 3D space intersect along a common straight line, how many solutions does the system of three linear equations have?
Answer: C
When three planes intersect along a line, every point on that line is a common solution, resulting in infinitely many solutions.
4. Solve the linear system: x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2. Find the value of x.
Answer: A
From Eq 1 + Eq 3: (x+y+z) + (x+2y-z) = 6 + 2 => 2x + 3y = 8. From Eq 1 + Eq 2: 3x + 2z = 9... Solving the system algebraically yields x = 1, y = 2, z = 3.
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 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.
7. 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.
8. Given x + 2y - z = 4, 2x - y + z = 3, and 3x + y + 2z = 11, calculate the value of z.
Answer: B
Solving this system gives x = 1, y = 2, z = 2.
9. 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.
10. In a system of three linear equations, if elimination leads to the statement 0 = 5, what does this indicate about the system?
Answer: C
An impossible identity such as 0 = 5 indicates a contradiction, meaning the planes do not share any common point of intersection, so there is no solution.