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Systems of Equations · Guided Practice

Systems of Equations

11 questions with answers and video solutions. Try each one before revealing the answer.

1
Use Gaussian elimination to solve this system of equations: 2x+y+z=2\displaystyle 2x+y+z=2, x−3y−z=5\displaystyle x-3y-z=5, x+y+2z=3\displaystyle x+y+2z=3.
Video solution coming soon
2
The points (1,−4)\displaystyle (1,-4), (2,−2)\displaystyle (2,-2) and (3,10)\displaystyle (3,10) lie on a parabola.
Find the equation of the parabola.
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3
Use Gaussian elimination to show that this system of equations involves redundancy (x+y+z=4\displaystyle x+y+z=4, 3x−y+2z=13\displaystyle 3x-y+2z=13, 2x−2y+z=9\displaystyle 2x-2y+z=9).
Obtain a parametric solution of the system of equations.
Video solution coming soon
4
Use Gaussian elimination to determine the value of k\displaystyle k which leads to redundancy in this system of equations (3x−y+z=2\displaystyle 3x-y+z=2, x+2y+2z=6\displaystyle x+2y+2z=6, x−5y+kz=−10\displaystyle x-5y+kz=-10).
Video solution coming soon
5
Use Gaussian elimination on the system of equations below to give an expression for z\displaystyle z in terms of λ\displaystyle \lambda (x+2y+6z=5\displaystyle x+2y+6z=5, x−4y−2z=1\displaystyle x-4y-2z=1, x−y+λz=−3\displaystyle x-y+\lambda z=-3).
For what value of λ\displaystyle \lambda is this system of equations inconsistent?
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6
Is the following system of equations ill-conditioned?
Explain your answer (10x+9y=5\displaystyle 10x+9y=5, 9x+8y=4\displaystyle 9x+8y=4).
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7
Is the following system of equations ill-conditioned?
Explain your answer (300x−y=−1\displaystyle 300x-y=-1, 299x−y=−2\displaystyle 299x-y=-2).
Video solution coming soon
8
2016 Q4
4 Marks

Below is a system of equations:

x+2y+3z=3\displaystyle x + 2y + 3z = 3
2x−y+4z=5\displaystyle 2x - y + 4z = 5
x−3y+2λz=2\displaystyle x - 3y + 2\lambda z = 2

Use Gaussian elimination to find the value of λ\displaystyle \lambda which leads to redundancy.

Video solution coming soon
9
2023 P1 Q3
3 Marks

A system of equations is defined by
x−3y+z=−1\displaystyle x - 3y + z = -1
3x−2y+4z=11\displaystyle 3x - 2y + 4z = 11
x+4y+2z=15\displaystyle x + 4y + 2z = 15
Use Gaussian elimination to determine whether the system shows redundancy, inconsistency or has a unique solution.

10
2024 P2 Q3
(4, 1, 1)6 Marks

(a)Use Gaussian elimination to express z\displaystyle z in terms of λ\displaystyle \lambda for the system of equations:

x−y−3z=1\displaystyle x - y - 3z = 1
2x−3y−5z=8\displaystyle 2x - 3y - 5z = 8
x+2y+λz=−7\displaystyle x + 2y + \lambda z = -7

(b)State the value of λ\displaystyle \lambda for which this system is inconsistent.

(c)Determine the solution of this system when λ=−1.\displaystyle \lambda = -1.

11
2026 P1 Q2
4 Marks

A system of equations is given by

x+y−z=9\displaystyle x+y-z=9
2x−y+3z=−2\displaystyle 2x-y+3z=-2
3x+2y−2z=21\displaystyle 3x+2y-2z=21

Use Gaussian elimination to solve this system of equations.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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