Solve equations with variables and constants on both sides
Pre-algebra students in OpenStax Prealgebra 2E solve equations with variables and constants on both sides using a four-step process: choose one side for variables and the other for constants, collect variable terms using the Addition or Subtraction Property of Equality, collect constants on the other side, then divide. For 9x + 3 = 4x + 23: subtract 4x from both sides to get 5x + 3 = 23, subtract 3 to get 5x = 20, divide to get x = 4. This general method handles integer, decimal, and fraction coefficients.
Key Concepts
Property Step 1. Choose one side to be the variable side and then the other will be the constant side. Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality. Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality. Step 5. Check the solution by substituting it into the original equation.
It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.
Examples Given $9x + 3 = 4x + 23$, subtract $4x$ from both sides to get $5x + 3 = 23$. Then subtract 3 to get $5x = 20$, so $x = 4$.
Common Questions
What are the steps for solving equations with variables on both sides?
1) Choose a variable side. 2) Move all variable terms there. 3) Move all constants to the other side. 4) Divide by the coefficient to solve.
How do you solve 9x + 3 = 4x + 23?
Subtract 4x: 5x + 3 = 23. Subtract 3: 5x = 20. Divide by 5: x = 4.
How do you solve (5/3)y + 2 = (2/3)y minus 4?
Subtract (2/3)y from both sides: y + 2 = -4. Subtract 2: y = -6.
Which side should you move variable terms to?
Move variables to the side with the larger coefficient to keep the coefficient positive, making the final division simpler.
What properties justify moving terms between sides?
The Addition and Subtraction Properties of Equality: adding or subtracting the same value from both sides keeps the equation balanced.