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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