Questions and Answers
For the system, use the second equation to make a substitution for y in the first equation (2x + y = 6, y = 3x + 4). What is the resulting equation?
2x + (3x + 4) = 6
For the system, use the second equation to make a substitution for x in the first equation (x + 5y - 10 = 0, x = 2y - 8). What is the resulting equation in simplest form?
7y - 18 = 0
Solve the following system of equations by the substitution method: 10x + 10y = 1, x = y - 3. What is the value of y?
31/20
For the system, use the second equation to make a substitution for x in the first equation (3x + 2y = 7, x - y + 3 = 0). What is the resulting equation?
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Solve the following system of equations (2x + y = 3, x = 2y - 1). Make sure there are NO SPACES in your answer. Include a comma in your answer.
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Which of the following equations could be the result of using the comparison method to solve the system (x + 2y = 6, x - 4y = 8)?
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Which of the following equations could be the result of using the comparison method to solve the system (x - 4y - 1 = 0, x + 5y - 4 = 0)?
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Based on the lesson, which of the following would be the best approach for solving the system (5x = y + 6, 2x - 3y = 4) by substitution?
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Solve the following system of equations by the substitution method (x - y = 0, x - y - 2 = 0). What is the solution set?
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In the system x - y = 4, x + y = 8, what is the x-coordinate of the solution to the system shown?
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In the system 2x + y = -2, x + y = 5, what is the x-coordinate of the solution to the system shown?
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When the second equation is subtracted from the first in the system 3x - 4y = 7, 3x + 2y = -5, what is the resulting equation?
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In the system x + y = k, x - y = k, what is the solution to the system shown?
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In the system 3x - 4y = 6, 6x - 8y = 10, how many solution(s) does the system shown have?
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In the system 3x + 5y = 78, 2x - y = 0, what is the x-coordinate of the point of intersection of the lines?
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In the system x + 5y = -2, 2x + y = 5, what is the y-coordinate of the point of intersection of the lines?
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Which of the following equations could be the result of multiplication and addition to eliminate a variable in the system (2x + 3y = 6, 5x + 2y = 4)?
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In the system 7x - 3y = 42, x - 4y = 1, what is the solution to the system of equations?
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The determinant found when column 1 consists of the x-coefficients and column 2 consists of the y-coefficients of a linear system is called what?
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What is the value of: (row 1, column 1)(row 2, column 2) - (row 1, column 2)(row 2, column 1) in a 2 by 2 matrix?
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What is a rectangular array made up of rows and columns called?
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The determinant found when column 1 consists of the constants and column 2 consists of the y-coefficients of a linear system is called what?
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What is the term for the constant preceding the variables in a product?
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What is the form Ax + By = C of a linear equation, where A, B, and C are integers called?
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The determinant found when column 1 consists of the x-coefficients and column 2 consists of the constants of a linear system is called what?
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Find the value for the following determinant: 2 3, 1 4.
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Find the value for the following determinant: 2 4, 3 9.
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Given the system of equations, what is the value of the system determinant? (x + y = 8, x - y = 10)
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Given the system of equations, what is the x-coordinate of the solution? (y = 3 - x, 3x + 4y = 1)
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Given the system of equations, what is the y-coordinate of the solution? (5x - 4y = 7, x = 5 - y)
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Given the system of equations, what is the solution? (3x - 2y + 10 = 0, 5y = 4x + 8)
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The Multiplicative Property of Equality states that for real numbers a, b, c, and d, if a = b and c = d, then ac = bd.
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Which of the following systems is equivalent to the given system (2/3x - 1/2y = 3, 1/2x + y = 5)?
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In the system (y = 1/4x + 7, y = 1/2x + 5), what is the y-coordinate of the solution?
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Which of the following equations is equivalent to x - y = 8?
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Which of the following systems is equivalent to the given system (0.01x - 0.3y = 1, y = 0.1x - 2)?
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Given the system (0.01x - 0.3y = 1, y = 0.1x - 2), what is the solution of the given system?
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Study Notes
Substitution Method
- The substitution method involves replacing a variable in one equation with an expression from another equation.
- E.g., from the system (2x + y = 6) and (y = 3x + 4), substituting yields (2x + (3x + 4) = 6).
- To solve (10x + 10y = 1) with (x = y - 3), results in (y = 31/20).
Simplifying Equations
- For the system (x + 5y - 10 = 0) with (x = 2y - 8), simplification leads to (7y - 18 = 0).
- Resultant equations may require reorganization for clarity.
Solution Sets
- A system may have a unique solution, no solution, or infinitely many solutions.
- The equation (x - y = 0) and (x - y - 2 = 0) leads to an empty solution set (Ø).
Determinants and Matrices
- The system determinant relates to the coefficients of a system of equations, denoting system solvability.
- Determinants calculate using the formula: (ad - bc) for a 2x2 matrix, producing insights into solutions.
- Various determinants include system, x-determinant, and y-determinant, tailored to the specific coefficients of equations.
Identifying Solutions
- The solution point of a system defines the intersection of its graph representations.
- Systems like (7x - 3y = 42) and (x - 4y = 1) results in ((13/22, 1/22)).
- The x-coordinate from (x + y = 8) and (x - y = 10) calculates to -2, denoting solution characteristics.
Graphical Intersections
- Equations such as (5x = y + 6) and (2x - 3y = 4) prompt methodical use of substitution to simplify.
- The point of intersection illustrates the relationship between two lines, with coordinates defined by solving systems algebraically.
Equivalent Systems
- Equivalent systems maintain solution consistency despite varying representation.
- Transforming systems, like from (0.01x - 0.3y = 1) to (x - 30y = 100), preserves solutions while adjusting coefficients.
Algebraic Properties
- The Multiplicative Property of Equality confirms that multiplying equal values maintains equivalence.
- Understanding coefficients and constants in equations leads to clearer analyses and problem-solving techniques.
Potential Solution Values
- Important solutions result from contextual understanding, like evaluating y-coordinates in (y = 1/4x + 7) and (y = 1/2x + 5) yielding precise results, such as a y-coordinate of 9.
- Solve completely defined systems to ensure a thorough grasp of algebraic manipulation and solution detection.
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