Solutions and Elementary Operations for Systems of Linear Equations
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Questions and Answers

What is the general form of a linear equation in R2?

  • ax + by = c (correct)
  • ax + by + cz = d
  • ax^2 + by^2 = c
  • ax + by + c = 0

    • What is the constant term in a linear equation with n variables?

  • The last variable in the equation
  • The sum of all the coefficients
  • The term not associated with any variable (correct)
  • The coefficient of the first variable

    • When is a tuple (s1, s2,...,sn) considered a solution to a linear equation a1x1 + a2x2 +...+ anxn = b?

  • When (s1, s2,..., sn) are all equal to 0
  • When a1s1 + a2s2 +...+ ansn = b (correct)
  • When s1, s2,..., sn are all prime numbers
  • When s1 = a1, s2 = a2,..., sn = an and b = 0

    • What is the possible outcome for a system of linear equations?

    <p>It may have no solution, a unique solution, or an infinite family of solutions</p> Signup and view all the answers

    What is a system called if it has no solution?

    <p>Inconsistent</p> Signup and view all the answers

    How many solutions does the system x + y + z = 4, x - y + z = 3 have?

    <p>Infinitely many solutions</p> Signup and view all the answers

    What are the solutions to x + y + z = 0 and x - y + z = 1 for any r ∈ R?

    <p>x = 1/2 - r, y = -1/2, z = r</p> Signup and view all the answers

    What is a matrix called that can be thought of as a table with real number entries?

    <p>Augmented matrix</p> Signup and view all the answers

    What are two systems of linear equations called if they have the same set of solutions?

    <p>Equivalent</p> Signup and view all the answers

    What is the general form of a linear equation in R3?

    <p>$ax + by + cz = d$</p> Signup and view all the answers

    In a linear equation with n variables, what are a1, a2,..., an known as?

    <p>Coefficients</p> Signup and view all the answers

    What is the solution to the system of linear equations x + y + z = 4 and x - y + z = 3?

    <p>(2, 2, 0)</p> Signup and view all the answers

    When is a system of linear equations said to have no solution?

    <p>When it contradicts itself</p> Signup and view all the answers

    Which term is used to describe a system of linear equations in R2 that has at least one solution?

    <p>Consistent</p> Signup and view all the answers

    For the system of equations x + y + z = 4 and x - y + z = 3, what is the number of solutions it has?

    <p>Infinitely many</p> Signup and view all the answers

    What is the set of solutions to the system x + y + z = 0 and x - y + z = 1 for any r ∈ R called?

    <p>Parametric solutions</p> Signup and view all the answers

    In the context of matrices, what is the matrix associated with a system of linear equations called?

    <p>Augmented matrix</p> Signup and view all the answers

    If two systems of linear equations are equivalent, what can be said about their augmented matrices?

    <p>They are identical</p> Signup and view all the answers

    What is the term for two straight lines in R2 that do not intersect?

    <p>Parallel</p> Signup and view all the answers

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