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Simultaneous Equations Graphical Solutions Solution by Addition or Subtraction Solution by Substitution Determinants Graphical Solutions Two linear equations plotted on the same coordinate axes will intersect at a single point 2x - 3y = 2 x + 2y = 8 Lines intersect at one point only: Exactly one solution x=4, y=2 Unless: The graphs are parallel The graphs are co-linear 4x + 6y = 12 2x + 3y = -6 Lines are parallel: No solution 2x – 3y = -6 -x + 3/2 y = 3 Lines coincide: Infinite solutions Graphical Solutions (2) Solve: x y 7 4 3 12 x y 1 2 4 4 Solution by Addition or Subtraction Put equations in the form (1) a1x + b1y = c1 (2) a2x + b2y= c2 Multiply the equations values that will produce the same coefficient for x or y. Add or subtract the equations to remove one of the variables Solve: 7 + y = 3x – 3 5–x=2–y Solution by Substitution Put equations in the form y = a1x + b1 or x = a1y + b1 Replace one variable with its equivalent expression Solve for the remaining variable Solve y = 0.4x y = 50 + .3x Determinants A determinant is the value computed from a square matrix of numbers by a rule of combining products of the matrix entries a b c d = ad -bc The number of rows (or columns) is called the order of the determinant. The diagonal, from the upper left corner to the lower right corner, is called the principal diagonal. The diagonal from the lower left corner to the upper right corner is called the secondary diagonal. Determinants (2) Using Determinants to solve linear equations Put equations in the form (1) a1x + b1y = k1 (2) a2x + b2y = k2 The denominator for both unknowns is: a1b2 – a2b1 The numerator of x is k1b2 – k2b1 The numerator of y is a1k2 – a2k1