Solving Three Simultaneous Equations
Simultaneous equations with three unknowns require a strategic approach, but once you master the technique, you'll find it's quite straightforward. The key is systematically eliminating variables one by one.
Let's work through this example: x + y + z = 6 ①, 2x - y + z = 3 ②, and 3x + y - 2z = -1 ③. Start by adding equations to eliminate one variable - here we'll get rid of y first.
Adding equations ① and ② gives us 3x + 2z = 9 ④. Then adding equations ② and ③ gives us 5x - z = 2 ⑤. Notice how we now have two equations with just x and z!
To solve these remaining equations, multiply equation ⑤ by 2 to get 10x - 2z = 4 ⑥. Adding equations ④ and ⑥ eliminates z completely: 13x = 13, so x = 1. Substitute back to find z = 3, then use either original equation to find y = 2.
Top Tip: Always check your answer by substituting all three values back into each original equation - they should all balance perfectly!