Solving Systems of Linear Equations in Three Variables
Solving Systems of Linear Equations in Three Variables
Warm-Up No Solution Infinitely many solutions
Here is a system of three linear equations in three variables: The ordered triple (2, -1, 1) is a solution to this system since it is a solution to all three equations.
The graph of a linear equation in three variables is a plane. Three planes in space can intersect in different ways The planes could intersect in a single point. The system has exactly one solution The planes could intersect in a line. The system has infinitely many solutions The planes could have NO point of intersection. The left figure shows planes that intersect pairwise, but all 3 do not have a common point of intersection. The right figure shows parallel planes. Each system has NO solution.
Substitution Method - Example 1 Since x+y=z, substitute this for z in the first two equations Simplify Finally, solve this linear system of two equations and two variables to get x = 4 and y =8 Since z=x+y, z = 12. Our final solution is (4, 8, 12)
Example 2 Solving for x in the first equation, we get 2 z + 10 =x. Substitute this for x in the last two equations -3(2 z+10) + y – 4 z = 20 -4(2 z+10) + 2 y + 3 z = -15 Simplify y – 10 z = 50 2 y -5 z = 25 Finally, solve this linear system of two equations and two variables to get y= 0 and y = -5 Since x=2 z + 10, x = 0. Our final solution is (0, 0, -5)
Example 3 Since t + 2 = s, substitute this for s in the first two equations 5 r +4(t+2) – 6 t = -24 -2(t+2) + 2 t = 0 Simplify -2 t - 4 + 2 t = 0 -4 = 0 Since we have a false statement, this system is inconsistent. Our final solution is No Solution
Try it on your own! A. B.
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