SIMULTANEOUS EQUATIONS - Nayland College
SIMULTANEOUS EQUATIONS - GEOMETRICAL INTERPRETATIONWe know that straight lines can either meet:1. At a point2. Be parallel3. Be the same line on top of each otherUNIQUE – 1 SOLNINCONSISTENT – 0 SOLNDEPENDENT – INFINITE SOLNSIn a similar way planes can either meet1. At a pointUNIQUE 1 solution2. In a line DEPENDENT infinite solutions(solved on calculator)2.DEPENDENT SOLUTION - three planes meet in a lineEach equation is a linear combination of the other two.Eg: 3 times eqn1 + 2 times eqn2 = eqn3 (for x, y, z and constant)How do we find what we multiply by:ie: a ()Looking at the x’s a + 2b = 7 Solve the simultaneous eqn to bLooking at the y’s 2a + b = 8 get a = 3 , b = 2 3. INCONSISTENT SOLUTION The three planes don’t intersect – think parallel planes3 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) Eg: eqn1 + 2 times eqn2 = eqn3 (for x, y, z BUT NOT FOR CONSTANT)How do we find what we multiply by:ie: a ()Looking at the x’s 4a + b = 6 Solve the simultaneous eqn bLooking at the y’s -a + 3b = 5 to get a = 1 , b = 2 2. All three planes parallel - coefficients of all 3 equations same or multiples of each, but not constantCoefficients eqn 1 = coefficients eqn 2 Coefficients eqn 3 are 2 times coefficients eqn 1but not constants3. Two planes parallel - coefficients of 2 equations same or multiples of each other, but not constantcoefficients eqn 3 are 4 times coefficients eqn1 but not constants ................
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