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What is a system of equations with 3 variables?

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A system of equations with 3 variables consists of three equations that involve three unknowns (usually x, y, and z). The solution is the set of values that satisfy all three equations simultaneously.

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How do you represent a system of equations using matrices?

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A system of equations can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.

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What is the determinant of a matrix?

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The determinant is a scalar value that is a function of the entries of a square matrix. It provides important information about the matrix, such as whether it is invertible.

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How do you calculate the determinant of a 3x3 matrix?

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For a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]], the determinant is calculated as: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).

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What does it mean if the determinant of a matrix is zero?

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If the determinant of a matrix is zero, it indicates that the matrix is singular, meaning it does not have an inverse and the system of equations may have either no solution or infinitely many solutions.

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What is the substitution method in solving systems of equations?

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The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equations.

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What is the elimination method in solving systems of equations?

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The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variables.

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How do you find the value of x in a system of equations?

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To find the value of x, you can use either the substitution or elimination method to isolate x in one of the equations.

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What is the significance of finding x - z in a system of equations?

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Finding x - z helps in understanding the relationship between the variables and can simplify the solution process.

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What is a consistent system of equations?

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A consistent system of equations has at least one solution, meaning the equations intersect at one or more points.