8.3 Solve Equations with Variables and Constants on Both Sides

Solving equations with variables and constants on both sides is a crucial skill in algebra. It involves isolating variables, combining like terms, and performing operations on both sides of the equation to find solutions.

This process builds on basic equation-solving techniques, extending them to more complex scenarios. By mastering these methods, you'll be able to tackle a wide range of algebraic problems and develop a strong foundation for advanced math concepts.

Solving Equations with Variables and Constants on Both Sides

Equations with constants on both sides

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• Isolate the variable on one side by performing the same operation on both sides of the equation
  • Subtract a constant from both sides if it is being added to the variable ($x + 3 = 7$ becomes $x = 7 - 3$)
  • Add a constant to both sides if it is being subtracted from the variable ($x - 4 = 2$ becomes $x = 2 + 4$)
  • Divide both sides by a constant if it is multiplying the variable ($3x = 15$ becomes $x = 15 ÷ 3$)
  • Multiply both sides by a constant if it is dividing the variable ($x ÷ 2 = 6$ becomes $x = 6 × 2$)
• Simplify the equation after performing the same operation on both sides to solve for the variable ($x + 2 = 8$ becomes $x = 6$)
• The process of moving terms from one side of the equation to the other is called transposition

Equations with variables on both sides

• Get all variables on one side and all constants on the other side by choosing one side for the variables and performing the opposite operation on both sides
  • Subtract a variable from both sides if it is being added on one side ($2x + 3 = x + 7$ becomes $x + 3 = 7$)
  • Add a variable to both sides if it is being subtracted on one side ($3x - y = 5$ becomes $3x = 5 + y$)
• Combine like terms on each side of the equation after getting all variables on one side ($2x - x + 3 = 7$ becomes $x + 3 = 7$)
• Isolate the variable by performing the appropriate operation with the constants on the other side ($x + 3 = 7$ becomes $x = 4$)

Strategy for complex equation solving

1. Simplify each side of the equation by combining like terms ($3x - 2 + 4x = 2x + 3 + 5x - 1$ becomes $7x - 2 = 7x + 2$)
2. Get all variables on one side of the equation by performing the appropriate operation on both sides ($7x - 2 = 7x + 2$ becomes $-2 = 2$)
3. Get all constants on the other side of the equation by performing the appropriate operation on both sides ($-2 = 2$ remains $-2 = 2$)
4. Isolate the variable by performing the appropriate operation with the constants ($-2 = 2$ becomes $2 = -2$ by dividing both sides by -1)
5. Determine the solution set, which includes all values that satisfy the equation

Like terms in linear equations

• Combine terms that have the same variable raised to the same power by adding or subtracting their coefficients ($3x + 2y - x + 4y$ becomes $2x + 6y$)
  • $3x$ and $-x$ are like terms with coefficients 3 and -1
  • $2y$ and $4y$ are like terms with coefficients 2 and 4
• Simplify complex linear equations by combining like terms on each side of the equation before solving for the variable ($2x + 3y - x = 5x - 2y + 1$ becomes $x + 3y = 4x - 2y + 1$)

Algebraic Expressions and Identities

• An algebraic expression is a combination of variables, numbers, and operations (e.g., 2x + 3y)
• An identity is an equation that is true for all values of the variables involved
• Algebraic expressions can be simplified by combining like terms and applying properties of operations
• Identities can be used to simplify complex equations and solve problems more efficiently