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Systems of Linear Equations (DP IB Analysis & Approaches (AA))

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  • What is a system of linear equations?

    A system of linear equations is a collection of two or more linear equations that involve the same variables.

    They are also called simultaneous equations.

  • How many linear equations are needed to solve a system involving n variables?

    n linear equations are needed to solve a system involving n variables.

  • What does a 2x2 system of linear equations represent geometrically?

    In a 2x2 system, each equation represents a straight line in 2D.

    The solution (if it exists and is unique) corresponds to the coordinates of the point where the two lines intersect.

  • What does a 3x3 system of linear equations represent geometrically?

    In a 3x3 system, each equation represents a plane in 3D.

    The solution (if it exists and is unique) corresponds to the coordinates of the point where the three planes intersect.

  • What form should equations be written in for GDC input in a 2x2 system?

    For a 2x2 system, equations often need to be written in the form a x plus b y equals c for GDC input.

    Make sure you know what form your GDC requires you to input equations in.

  • What is an important step to take when solving a system of linear equations on a non-calculator paper?

    On a non-calculator paper, it's important to check your final answer by inputting the values into all original equations to ensure that they satisfy the equations.

  • How can a system of linear equations be rewritten in a shorter way than writing out the full equations?

    E.g. a subscript 1 x plus b subscript 1 y equals c subscript 1 a subscript 2 x plus b subscript 2 y equals c subscript 2

    A system of linear equations can be rewritten using just the coefficients and not the variables.

    E.g. a subscript 1 x plus b subscript 1 y equals c subscript 1 a subscript 2 x plus b subscript 2 y equals c subscript 2 becomes open square brackets table row cell right enclose table row cell table row cell table row cell a subscript 1 end cell cell space b subscript 1 end cell end table end cell end table end cell row cell table row cell a subscript 2 end cell cell space b subscript 2 end cell end table end cell end table end enclose end cell cell table row cell c subscript 1 end cell row cell c subscript 2 end cell end table end cell end table close square brackets

  • What is row reduction, in the context of linear systems of equations?

    Row reduction is a sequence of operations on successive rows that simplify the matrix of coefficients.

  • True or False?

    Row operations change the solution of a system of linear equations.

    False.

    Row operations do not change the solution of a system of linear equations.

  • What is the row reduced form in a system of linear equations?

    The row reduced form is a system of linear equations where the coefficient matrix has 1s in the leading diagonal and zeros below them.

  • What are the three types of row operations in a system of linear equations?

    The three types of row operations in a system of linear equations are:

    1. Switching rows.

    2. Multiplying a row by a constant.

    3. Adding multiples of one row to another.

  • What is the first step in row reducing a system of linear equations?

    The first step in row reducing a system of linear equations is to get a 1 in the top left corner by dividing the first row by its leading coefficient.

  • How do you create zeros below a leading 1 in row reduction in a system of linear equations?

    To create zeros below a leading 1 in row reduction in a system of linear equations, add or subtract multiples of the row with the leading 1 from the rows below it.

  • True or False?

    In row reduction, you must always make every equation have a single variable with a coefficient of 1.

    False.

    In row reduction, you only need to make one equation have a single variable with a coefficient of 1.

    You can then use that use that result to work out the others.

  • How many possible types of solutions can a system of linear equations have?

    A system of linear equations can have three possible types of solutions:

    • one unique solution,

    • no solutions,

    • or an infinite number of solutions.

  • What is meant by an inconsistent system, in the context of linear systems of equations?

    An inconsistent system is a system of linear equations that has no solutions.

  • How can you identify if a system has no solutions from its row reduced form?

    A system has no solutions if its row reduced form contains at least one row where the entries to the left of the line are zero and the entry on the right of the line is non-zero.

    For example if D subscript 2 is non-zero in this matrix: stretchy left square bracket table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 0 0 row 0 0 1 end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row cell D subscript 3 end cell end table stretchy right square bracket.

  • What indicates that a system has an infinite number of solutions in its row reduced form?

    A system has an infinite number of solutions if its row reduced form contains at least one row where all entries are zero and there are no inconsistent rows.

    For example: open square brackets table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 1 cell C subscript 2 end cell row 0 0 0 end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row 0 end table close square brackets.

  • What is a dependent system, in the context of linear systems of equations?

    A dependent system of linear equations is one that has an infinite number of solutions.

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      Further Complex Numbers

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      Systems of Linear Equations

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      Functions Toolkit

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