Solve a system of equations in three variables using elimination

Learning Objectives Identify a system of three linear equations and its geometric representation as intersecting planes. Strategically select a variable to eliminate from a system of three equations. Systematically reduce a 3x3 system of equations into a 2x2 system by performing elimination twice. Solve the resulting 2x2 system for two of the variables. Use back-substitution to find the value of the third variable. Verify the solution by checking the ordered triple in all three original equations. How can you find the exact point in 3D space where three different planes intersect? ✈️ Solving systems of three equations gives us the coordinates of that unique point! This tutorial will guide you through the elimination method, a powerful and systematic algebraic technique to s...

TermDefinitionExample System of Three Linear EquationsA set of three linear equations that share the same three variables (e.g., x, y, and z). Geometrically, this represents three planes in a three-dimensional coordinate system.1) x + 2y - z = 4 2) 2x - y + 3z = 7 3) 3x + y + 2z = 8 Ordered TripleA set of three numbers (x, y, z) that represents a point in three-dimensional space. It is the solution to a system of three equations.The ordered triple (1, 2, 1) is a potential solution to a system with variables x, y, and z. Elimination MethodAn algebraic method for solving a system of equations where you add or subtract equations (or multiples of equations) in order to cancel out one of the variables.Adding the equations (x + y = 5) and (x - y = 1) results in 2x = 6, which eliminates the &#03...

Addition Property of Equality If A = B and C = D, then A + C = B + D This property allows us to add two equations together. Since each side of an equation is equal, adding them to another equation maintains the equality, helping to eliminate a variable with opposite coefficients. Multiplication Property of Equality If A = B, then kA = kB for any non-zero constant k This property allows us to multiply an entire equation by a constant. We use this to create opposite coefficients for a variable we want to eliminate before adding the equations.