Section 1
Coefficient Matching for Linear Expressions
Property
If two linear expressions are equal for all values of a variable, then their corresponding coefficients must be equal:
If for all values of , then and
In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to solve systems of three-variable linear equations using substitution and elimination to reduce them to simpler two-variable systems. Working through problems like solving for x, y, and z simultaneously, students apply techniques such as multiplying equations to match coefficients and strategically eliminating variables step by step. This lesson is part of Chapter 5 on Multi-Variable Linear Equations and also introduces setting up multi-variable equations from word problems involving unknown quantities.
Section 1
Coefficient Matching for Linear Expressions
If two linear expressions are equal for all values of a variable, then their corresponding coefficients must be equal:
If for all values of , then and
Section 2
Solving 3x3 Systems by Substitution
To solve a system of three equations with three variables using substitution, follow these steps:
From the first equation, we can isolate : . Substitute this into the other two equations to get a 2x2 system in terms of and .
Substitute into the second equation: , which simplifies to . Now you have a 2x2 system with the equations and , which indicates no solution.
The substitution method systematically reduces a complex problem into a simpler one. By solving for one variable and substituting its expression into the other equations, you reduce a three-variable system to a familiar two-variable system. Once you solve the simpler system, you can work backwards to find the value of the initial variable. This strategy is particularly effective when one of the equations can be easily rearranged to isolate a variable.
Section 3
Gaussian reduction for three-variable systems
Gaussian reduction is a systematic method for solving three-variable linear systems by using elimination to create a triangular form, then applying back-substitution.
Steps for Gaussian Reduction on a System:
Expand to review the lesson summary and core properties.
Section 1
Coefficient Matching for Linear Expressions
If two linear expressions are equal for all values of a variable, then their corresponding coefficients must be equal:
If for all values of , then and
Section 2
Solving 3x3 Systems by Substitution
To solve a system of three equations with three variables using substitution, follow these steps:
From the first equation, we can isolate : . Substitute this into the other two equations to get a 2x2 system in terms of and .
Substitute into the second equation: , which simplifies to . Now you have a 2x2 system with the equations and , which indicates no solution.
The substitution method systematically reduces a complex problem into a simpler one. By solving for one variable and substituting its expression into the other equations, you reduce a three-variable system to a familiar two-variable system. Once you solve the simpler system, you can work backwards to find the value of the initial variable. This strategy is particularly effective when one of the equations can be easily rearranged to isolate a variable.
Section 3
Gaussian reduction for three-variable systems
Gaussian reduction is a systematic method for solving three-variable linear systems by using elimination to create a triangular form, then applying back-substitution.
Steps for Gaussian Reduction on a System: