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AS91587 Simultaneous Equations • In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. This is a system of linear equations This is not a system of linear equations 3x - 2y + z = 4 2x + y - z = 8 x- y+3 z = 2 x + y - z = 10 2x + 3y + z = 12 x - y + 5z = 1 • is a linear system of three equations in the three variables x, y, z. Solving equations 2x + 4 = x - 3 x = -7 • There is a unique solution as -7 is the only value of x that makes the LHS = RHS We don’t always get a solution though • If we try to solve the following: 2x - 3 = 2x + 7 0 = 10 • We find that this is not true and hence there are no solutions. Solving 2D systems • Understanding the equations: • Example: y = 2x+3 • This equations has 2 variables and there are an infinite number of solutions i.e. Every point on this line satisfies the equation so there are an infinite number of solutions. y = 2x + 3 e.g. (0, 3), (1,5), (0.2,3.4) are some of these solutions y = 2x + 3 We can only get a solution for ‘y’ if we know the particular value of ‘x’ i.e. ‘x’ is no longer a variable. y = 2x + 3 x=2 y = 2´2+3 =7 2 equations with 2 variables each • Solve y = 2x + 3 y = -x + 6 Solving using substitution • Solve y = 2x + 3 y = -x + 6 Þ -x + 6 = 2x + 3 1= x y=5 Solving using elimination y = 2x + 3 (1) y = -x + 6 (2) 0 = 3x - 3 (Subtracting (1)-(2)) x =1 y=5 There is only one point that lies on both lines and so this point (1, 5) is a unique solution Example • Solve: 2x + 3y = 11 x+y = 4 Example 2x + 3y = 11 (1) x + y = 4 (2)´ 2 2x + 2y = 8 (3) y = 3 (1) - (3) x =1 There is a unique solution (1, 3) If the lines have different gradients, they must intersect and give us a unique solution. Example • Solve: y - 2x = 3 y - 2x = -1 Example y - 2x = 3 y - 2x = -1 0 = 4 (subtracting) • This is not possible The lines will never intersect and hence there is no solution Notice that the LHS is the same but the RHS is different y - 2x = 3 y - 2x = -1 • This means the lines have the same gradient but are separated. But if the LHS is the same as the RHS, then every point matches and hence there are an infinite number of solutions y - 2x = 3 y - 2x = 3 • This means the lines have the same gradient but are separated. It could look like this y - 2x = 3 2y - 4x = 6 • The second line is a multiple of the first line Summary • Left hand sides have different gradients so we expect a unique solution 2x + 3y = 7 x - 5y = -3 The systems of equations are consistent with a unique solution Summary • Left hand sides have the same gradients and the right hand sides are different so we expect no solution 2x + 3y = 7 4x + 6y = -3 No solutions. The system of equations is inconsistent. Summary • Left hand sides have the same gradients and the right hand sides are in proportion so we expect infinite solutions 2x + 3y = 7 4x + 6y = 14 One line matches the other line exactly and so they have infinite solutions. The system of equations is consistent with infinite solutions