In this Grade 4 AoPS Introduction to Algebra lesson from Chapter 22, students learn how to identify symmetric expressions and symmetric systems of equations, then exploit that symmetry to solve multi-variable systems efficiently. Using techniques like summing all equations simultaneously and multiplying equations together, students find solutions to systems with four or more variables, as practiced through AMC-style problems involving products and sums of unknowns.
Section 1
Identifying Symmetric Expressions and Systems
A symmetric expression remains unchanged when any two variables are swapped. For variables , , , an expression is symmetric if swapping any pair (like ) produces an identical expression.
Section 2
Elimination with Symmetric Systems
To solve symmetric systems of equations by elimination:
Section 3
Adding Symmetric Equations to Find Variable Sums
In a symmetric system of equations, adding all equations together eliminates asymmetric terms and reveals the sum of all variables. For a system with variables , , , the sum can be found directly by adding the left and right sides of all equations.
Section 4
Product Method for Symmetric Systems
In symmetric systems, multiplying all equations together produces an equation containing the product of all variables. For a system with variables , , , the product equation can be used to find directly.
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Section 1
Identifying Symmetric Expressions and Systems
A symmetric expression remains unchanged when any two variables are swapped. For variables , , , an expression is symmetric if swapping any pair (like ) produces an identical expression.
Section 2
Elimination with Symmetric Systems
To solve symmetric systems of equations by elimination:
Section 3
Adding Symmetric Equations to Find Variable Sums
In a symmetric system of equations, adding all equations together eliminates asymmetric terms and reveals the sum of all variables. For a system with variables , , , the sum can be found directly by adding the left and right sides of all equations.
Section 4
Product Method for Symmetric Systems
In symmetric systems, multiplying all equations together produces an equation containing the product of all variables. For a system with variables , , , the product equation can be used to find directly.
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter