Questions and Answers
What does 'substitute' mean?
What are the steps for substitution in solving a linear system?
Substitute an equivalent expression for a variable, solve the resulting equation, substitute the value back into the original equation, solve for the other variable, check the solution.
What is the resulting equation in simplest form for the substitution x + 5y - 10 = 0, x = 2y - 8?
7y - 18 = 0
What is the value of y when solving the system 10x + 10y = 1, x = y - 3?
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What is the resulting equation when making a substitution for y in the equation 2x + y = 6, where y = 3x + 4?
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What is the resulting equation for the system 3x + 2y = 7, x - y + 3 = 0?
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For the system 8x = 2y + 5, 3x = y + 7, what is the solution set?
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How many birds and cats are counted by Shawna and Kai in the pet store?
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What are the equations to model the situation for the cell phone companies?
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What is a suitable approach for solving the system 5x = y + 6, 2x - 3y = 4?
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Study Notes
Substitution Method in Algebra
- Substitute: Replace a quantity with its equivalent expression in equations.
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Steps for Substitution:
- Substitute an equivalent expression for a variable into one equation.
- Solve for the other variable.
- Substitute that value back into one of the original equations.
- Solve for the remaining variable.
- Check the ordered pair in the other equation and state the solution set.
Application of Substitution Method
- To solve a linear system by substitution, isolate one variable, substitute into the other equation, and resolve.
- Use algebraic methods for accurate solutions in linear equations.
Example Problems
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Solve the system:
- 10x + 10y = 1
- x = y - 3
- Result: y = 31/20.
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Substitute:
- For the equations:
- x + 5y - 10 = 0
- x = 2y - 8
- Resulting equation: 7y - 18 = 0.
- For the equations:
Common Misunderstandings
- Identify incorrect outcomes from substitutions in given systems, e.g., from 2x + y = 7 and y - x = 1 producing incorrect equations.
Real-World Applications
- Pet Store Problem: Shawna counts 52 animals (birds and cats), while Kai counts 138 legs. Results: 32 birds, 17 cats.
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Cell Phone Cost Comparison: Model costs with equations:
- Company A: c = 20 + 0.03t
- Company B: c = 5 + 0.07t
- Costs equal at t = 2 texts.
Special Cases
- Systems may have no solutions or infinite solutions. Example: For the system x - y = 0 and x - y - 2 = 0, the solution set is empty (∅).
- Solutions must be checked against both original equations to ensure accuracy.
Conclusion
- Solving systems graphically for non-integer solutions can be challenging; algebraic methods provide exact solutions.
- The substitution property is fundamental in solving linear equations efficiently.
- Solutions to linear systems are always expressed as ordered pairs.
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Master the substitution method in algebra with these flashcards. Each card highlights key definitions and steps to effectively substitute variables and solve equations. Perfect for students looking to enhance their understanding of algebraic concepts.
