1. Solve the following systems of simultaneous equations
|
a
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2y + 3x = 6 y = 4x − 2 |
b
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3x + 4y = 12 2x − 3y = 8 |
c
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3x − 2y + 3z = -16 x + 2y − z = 6 2x − y + 3z = -13 |
|
d
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x + y + z = 3 4x + 2y + z = 1 9x + 3y + z = 1 |
e
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x − z = 1 2x − y = 8 2y − 3z = -10 |
f
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3p + 5q + r = 3 15p + 15q + 2r = 12 30p + 20q + 2r = 21 |
2. Find the solution to the equations:
27x − 10y − 2z = -24
x + 5y − z = 18
6x + y − z = 6
Comment on your results.
3. Show by using the elimination method that the system given is inconsistent, with no solution.
x + 3y + z = 0
4x + 2y − z = 0
-x + y + z = 1
4. Consider the following system of two linear equations, where c is a constant:
2x + 5y = 16
4x + cy = 25
a. Solve the system when c = 3
b. Give a value of the constant c for which the system is inconsistent.
c. If c is chosen so that the system is consistent, explain in geometrical terms why there is a unique solution.
5. A man goes shopping and buys 5 cans of catfood, 2 packets of biscuits and 2 cans of fruit. This costs him $17.74.
On another occasion he buys 4 cans of cat food, 3 packets of biscuits and one can of fruit, at a cost of $15.44.
Later again, he buys 6 cans of cat food, 4 packets of biscuts and 3 cans of fruit for $25.80.
Use simultaneous equations to help find the cost of each item.
6. Solve the following simultaneous equations:
7. When newsprint is supplied to a newspaper company it is rewound from large reels to smaller reels. During this rewinding process the tension on the paper should not change. This is achieved by altering the load on the smaller reel as its diameter increases.
The relationship between the load, L kilonewtons per metre (L kNm-1), and the reel diameter, x metres, is modelled by a graph consisting of two parabolic arcs, AB and BC, as shown.
a. Arc AB is part of the parabola L = px2 + qx + r. Points D(0.1, 2.025). E(0.2, 2.9) and F(0.3, 3.425) lie on arc AB. Set up a system of three simultaneous equations relating p, q and r. Do not solve the equations.
b. Arc BC is part of the parabola L = sx2 + tx + u. The coefficients s, t and u can be found by solving the following system of linear equations.
0.25s + 0.5t + u = 3.5
0.36s + 0.6t + u = 3.2
0.64s + 0.8t + u = 2.0
Find s, t and u. It is necessary to show full algebraic working.
