Solving Systems of Linear Equations: Substitution

Questions and Answers

What is the value of y when the value of x in the system of equations is 1?

What can Vineet conclude if solving a system of equations resulted in the equation 7 = 9?

The system has a unique solution.

The system has infinitely many solutions.

The equations are equivalent.

The system of equations does not have a solution. (correct)

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.

10, 105 Signup and view all the answers

What is the equation represented by the second table if the first table has the equation y = 5.75x + 34.5?

y = -2.5x - 15 Signup and view all the answers

What is the value of x from the equations 2x + 3y = 3 and y = 8 - 3x?

1 Signup and view all the answers

What is the solution to the system of equations 3/8x + 1/3y = 17/24 and x + 7y = 8?

(1, 1) Signup and view all the answers

What system of equations can be used to find how many problems of each point value are on Mr. Martin's math test?

x + y = 29, 5x + 2y = 100 Signup and view all the answers

How many problems of each point value are on Mr. Martin's math test?

14 problems worth 5 points and 15 problems worth 2 points Signup and view all the answers

Who is correct when Jared substitutes x + 10 for y and Nicole substitutes y - 10 for x in their equations?

Both are correct. Signup and view all the answers

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.

Description

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!