This worksheet focuses on a crucial pre-algebra skill: manipulating equations before substitution. Mastering this technique significantly simplifies solving systems of equations and tackling more complex algebraic problems. We'll cover various methods and provide ample practice to solidify your understanding.
Why Manipulate Equations Before Substitution?
Substitution, a method for solving systems of equations, involves solving for one variable in terms of another and then substituting that expression into the second equation. However, sometimes the equations aren't conveniently set up for direct substitution. Manipulating the equations beforehand makes the substitution process much easier and less error-prone. This often involves isolating a variable or simplifying the equations.
Common Manipulation Techniques
Before substituting, consider these techniques:
1. Isolating a Variable: The most common manipulation is isolating a variable in one equation. This means getting one variable entirely by itself on one side of the equals sign. For example:
- Original Equation: 2x + y = 5
- Manipulated Equation: y = 5 - 2x (We isolated 'y')
2. Simplifying Equations: Sometimes, equations can be simplified before substitution. This could involve combining like terms, expanding brackets, or factoring. For instance:
- Original Equation: 3(x + 2) + y = 11
- Simplified Equation: 3x + 6 + y = 11 (Expanded the bracket)
- Further Simplified Equation: 3x + y = 5 (Combined like terms)
3. Rearranging Equations: Sometimes, it's beneficial to rearrange an equation to make substitution easier. This may involve moving terms from one side to the other while maintaining balance.
Practice Problems
Let's put these techniques into practice. For each problem, manipulate the given equations to make substitution easier, then solve the system. Show your work clearly, indicating each step of your manipulation.
Problem 1:
Solve the following system of equations using substitution after manipulating the equations:
- 2x + y = 7
- x - 3y = 4
Problem 2:
Solve the following system using substitution after manipulation:
- x + 2y = 10
- 3x - y = 9
Problem 3:
Solve the system after manipulating the equations:
- 2(x + 1) + y = 8
- 4x - y = 6
Problem 4 (Challenge):
Solve the following system after appropriate manipulation:
- 2x + 3(y - 1) = 7
- x - 2y = -1
Problem 5 (Challenge):
Solve the system:
- 0.5x + y = 3
- 2x - 0.2y = 4
Solutions (For Teacher/Self-Check)
(Remember to show your working for each problem to understand the process fully!)
Problem 1 Solution: Manipulate the first equation to isolate 'y': y = 7 - 2x. Substitute this into the second equation and solve for x, then substitute the x value back to find y.
Problem 2 Solution: Manipulate either equation to isolate a variable. Substituting this into the other equation will allow you to solve for one variable and subsequently the other.
Problem 3 Solution: Simplify the first equation before isolating a variable for substitution into the second equation.
Problem 4 Solution: Simplify the first equation before deciding which variable to isolate for substitution.
Problem 5 Solution: Work with decimals carefully. Consider multiplying equations by 10 to get rid of the decimals, simplifying calculations.
This worksheet provides a solid foundation for manipulating equations before substitution. Remember to practice consistently to build proficiency. Understanding these manipulation techniques is crucial for advanced algebra and other mathematical concepts.
