Solve a system of equations in three variables using substitution

Learning Objectives Isolate a single variable in one equation of a three-variable system. Substitute an algebraic expression into two other equations to create a new system of two equations in two variables. Solve the resulting two-variable system to find the values of two of the variables. Use back-substitution to find the value of the third, final variable. Write the final solution as an ordered triple (x, y, z). Verify the solution by plugging the ordered triple into all three original equations. Imagine you're an aerospace engineer calculating the precise trajectory for a satellite. You need to find the one point (x, y, z) where three different flight paths intersect. How do you find that exact coordinate? 🛰️ This tutorial will guide you through the substitution me...

TermDefinitionExample System of Three Linear EquationsA set of three linear equations that share the same three variables (e.g., x, y, and z). The solution is the single point where the three planes represented by the equations intersect.1) x + y + z = 6 2) 2x - y + z = 3 3) 3x + y - 2z = -1 Ordered TripleA set of three numbers written in the format (x, y, z) that represents the solution to a system in three variables. It corresponds to a specific point in three-dimensional space.The ordered triple (1, 2, 3) is the solution to the system in the previous example. Substitution MethodAn algebraic method for solving a system of equations by solving one equation for a variable and then substituting that resulting expression into the other equations.From x + y = 5, we can isolate x to get x = 5...

The Isolation Principle Given an equation Ax + By + Cz = D, isolate one variable. For example, to isolate x: x = (D - By - Cz) / A This is the first step of the substitution method. Choose one equation and one variable that is easy to solve for (ideally, one with a coefficient of 1 or -1 to avoid fractions). The Reduction Principle System 1: {Eq1, Eq2, Eq3} → System 2: {New EqA, New EqB} After isolating a variable from one equation (e.g., Eq1), substitute the resulting expression into the *other two* equations (Eq2 and Eq3). This reduces the 3x3 system into a 2x2 system with only two variables.