Solving Systems of Three Equations
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Questions and Answers

What is the primary goal of the first step in solving a system of three equations with three unknowns using elimination?

  • To simplify the original equations.
  • To solve for one of the variables directly.
  • To eliminate one of the variables. (correct)
  • To reduce the system to two equations with two unknowns.

    • When choosing a pair of equations in Step 2, which factor should be considered most important?

  • The consistency of the constant terms in the equations.
  • The simplicity of the coefficients of the equations.
  • The ease of eliminating one variable from the pair. (correct)
  • The presence of variables with opposite signs in the equations.

    • Which of the following statements accurately reflects the outcome of Step 3?

  • The eliminated variable is found, but the remaining two variables are still unknown.
  • A single equation involving two variables is obtained by combining the chosen equations. (correct)
  • Two equations remain, but one variable is eliminated from both equations.
  • The elimination process results in two new equations, each with only one variable.

    • What is the purpose of Step 4?

    <p>To create a second equation with only two variables, similar to the result of Step 3. (D)</p> Signup and view all the answers

    In Step 5, why is it necessary to solve for one variable in terms of the other?

    <p>To make it easier to substitute one variable into the other equation. (D)</p> Signup and view all the answers

    Why is back-substitution used in the final step of the process to find the third unknown?

    <p>To verify that the solution satisfies all three original equations. (C)</p> Signup and view all the answers

    Which of the following techniques is NOT commonly used to solve a system of three equations with three unknowns?

    <p>Quadratic formula. (A)</p> Signup and view all the answers

    If you have a system of three equations where two of the equations are identical after applying elimination, what can you conclude?

    <p>The system has infinitely many solutions. (A)</p> Signup and view all the answers

    Study Notes

    Solving Systems of Three Simultaneous Equations

    • Method: Substitution, elimination, or matrices (Gaussian elimination) are used to find solutions.

    • Example System:

      • 2x + y - 3z = 5
      • 3x - y + z = 4
      • x + 2y + z = 7

    Step-by-Step Solution (using Elimination):

    • Step 1: Write down the system.

    • Step 2: Choose two equations. Eliminate a variable. Here, work with equations (a) and (b) to eliminate 'y'.

    • Step 3: Eliminating 'y' through addition:

      • (2x + y - 3z) + (3x - y + z) = 5 + 4
      • Resulting equation: 5x - 2z = 9 (Equation 4)

    • Step 4: Choose another pair and eliminate the same variable. Eliminating 'y' from equations (b) and (c):

      • (3x - y + z) + (x + 2y + z) = 4 + 7
      • Resulting equation: 4x + y + 2z = 11 (Equation 5)

    • Step 5: Solve the resulting two equations (Equation 4 and 5) with two unknowns (x and z).

      • Use substitution or elimination methods to solve for x and z.

    • Step 6: Back-substitution: Substitute solved values of x and z into one of the original equations to find the value of y.

    Key Concepts

    • Elimination: Adding or subtracting equations to remove a variable.
    • Substitution: Replacing one variable with an expression of other variables.
    • Back-substitution: Substituting found values into an original equation to obtain the remaining variable.

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    Description

    Test your knowledge on solving systems of three simultaneous equations using methods such as substitution, elimination, and Gaussian elimination. This quiz will guide you through different steps to find the solutions effectively. Challenge yourself with practical examples and improve your problem-solving skills.

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