Questions and Answers
What is the value of y when the value of x in the system of equations is 1?
6
What can Vineet conclude if solving a system of equations resulted in the equation 7 = 9?
What is the linear equation that represents the price of the T-shirt company Shirt Box?
y = 7.5x + 30
For the total cost to be the same for both companies, the volleyball team would need to purchase ____ T-shirts from each company for a total of $___ for each company.
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What is the equation represented by the second table if the first table has the equation y = 5.75x + 34.5?
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What is the value of x from the equations 2x + 3y = 3 and y = 8 - 3x?
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What is the solution to the system of equations 3/8x + 1/3y = 17/24 and x + 7y = 8?
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What system of equations can be used to find how many problems of each point value are on Mr. Martin's math test?
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How many problems of each point value are on Mr. Martin's math test?
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Who is correct when Jared substitutes x + 10 for y and Nicole substitutes y - 10 for x in their equations?
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Study Notes
Solving Linear Equations: Substitution Method
- Value of x: In the system of equations 3x + y = 9 and y = -4x + 10, x is determined to be 1; substituting into the second equation gives y = 6.
- Conclusion of No Solution: If the substitution leads to an equation like 7 = 9, the system of equations has no solution, indicating inconsistency.
Price Comparison for T-Shirts
- Linear equations for T-shirt companies: For Shirt Box, y = 7.5x + 30; for Just Tees, y = 10.5x describes their pricing models.
- Equal Total Cost Calculation: For equal costs from both companies, the volleyball team must buy 10 T-shirts for $105 each.
Functions Representation and Solutions
- Linear function from tables: The first equation from the first table is y = 5.75x + 34.5; the second is y = -2.5x - 15.
- Solution of the system: The intersection point or solution of equations from the tables is (-6, 0).
System of Equations Example
- Used equations: For the system 2x + 3y = 3 and y = 8 - 3x, substitute y back into the first equation leading to 2x + 24 - 9x = 3 with resultant x = 3 and y = -1.
- Resulting Point: For the two variable equations 3/8x + 1/3y = 17/24 and x + 7y = 8, the solution is (1, 1).
Math Test Problems Composition
- Setting up the equations: Mr. Martin’s test includes a total of 29 problems with point values of 5 and 2; equations defined as x + y = 29 and 5x + 2y = 100.
- Number of Problems: The solution revealed is 14 problems worth 5 points and 15 problems worth 2 points, fulfilling the conditions set.
Substitution Method and Variable Manipulation
- Variable substitution: For y = x + 10 and x = y - 10, both Jared and Nicole accurately apply substitution, emphasizing the flexibility of solving for either variable using the appropriate expression.
- Correctness of Method: Both methods enable the resolution of linear systems, demonstrating the interchangeable nature of the variables when substituting correctly.
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Test your understanding of solving systems of linear equations using the substitution method with these flashcards. Each card presents a scenario where you have to substitute and solve for variables. Perfect for mastery of this essential algebraic technique!