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SIMULTANEOUS EQUATIONS - GEOMETRICAL INTERPRETATION
We know that straight lines can either meet:
1. At a point
2. Be parallel
3. Be the same line on top of each other
UNIQUE – 1 SOLN
INCONSISTENT – 0 SOLN
DEPENDENT – INFINITE SOLNS
In a similar way planes can either meet
1. At a point UNIQUE 1 solution
(solved on calculator)
2.
2. In a line DEPENDENT infinite solutions
DEPENDENT SOLUTION - three planes meet in a line
Each equation is a linear combination of the other two.
x + 2y + z = 4
Eg:
2x + y - 2z = 5
3 times eqn1 + 2 times eqn2 = eqn3 (for x, y, z and constant)
7x + 8y - z = 22
How do we find what we multiply by:
ie: a ( x + 2y + z = 4)
b (2x + y - 2z = 5)
7x + 8y - z = 22
Looking at the x’s a + 2b = 7
Looking at the y’s 2a + b = 8
Solve the simultaneous eqn to
get a = 3 , b = 2
3. INCONSISTENT SOLUTION
The three planes don’t intersect – think parallel planes
3 cases: - Hint (coefficients only – NOT the CONSTANTS)
1. . Each plane is parallel to the intersection of the other two planes – toblerone
(similar to Dependent but not for constant)
4 x - y + 2z = 5
Eg:
x + 3y - 5z = 4
eqn1 + 2 times eqn2 = eqn3 (for x, y, z BUT NOT FOR CONSTANT)
6x + 5y - 8z = 15
How do we find what we multiply by:
ie: a ( 4x - y + 2z = 5)
b (x + 3y - 5z = 4)
Looking at the x’s 4a + b = 6
Looking at the y’s -a + 3b = 5
Solve the simultaneous eqn
to get a = 1 , b = 2
6x + 5y - 8z =13
2.
All three planes parallel - coefficients of all 3 equations same or multiples of each, but not constant
x + 2y + 3z = 5
x + 2y + 3z = 7
2x + 4 y + 6z = 4
Coefficients eqn 1 = coefficients eqn 2
Coefficients eqn 3 are 2 times coefficients eqn 1
but not constants
3. Two planes parallel - coefficients of 2 equations same or multiples of each other, but not constant
x + 2y + z = 10
x + 3y + 2z = 6
4 x + 8y + 4z = 12
coefficients eqn 3 are 4 times coefficients eqn1
but not constants