Podcast
Questions and Answers
What is the primary goal of the first step in solving a system of three equations with three unknowns using elimination?
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?
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?
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?
What is the purpose of Step 4?
In Step 5, why is it necessary to solve for one variable in terms of the other?
In Step 5, why is it necessary to solve for one variable in terms of the other?
Why is back-substitution used in the final step of the process to find the third unknown?
Why is back-substitution used in the final step of the process to find the third unknown?
Which of the following techniques is NOT commonly used to solve a system of three equations with three unknowns?
Which of the following techniques is NOT commonly used to solve a system of three equations with three unknowns?
If you have a system of three equations where two of the equations are identical after applying elimination, what can you conclude?
If you have a system of three equations where two of the equations are identical after applying elimination, what can you conclude?
Flashcards
Simultaneous equations
Simultaneous equations
Equations that are solved together because they share common variables.
Methods to solve equations
Methods to solve equations
Common methods include substitution, elimination, and matrices (like Gaussian elimination).
Elimination method
Elimination method
A technique to remove a variable by adding or subtracting equations.
Substitution method
Substitution method
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Coefficient
Coefficient
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Back-substitution
Back-substitution
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Gaussian elimination
Gaussian elimination
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Two unknowns
Two unknowns
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Study Notes
Solving Systems of Three Simultaneous Equations
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Method: Substitution, elimination, or matrices (Gaussian elimination) are used to find solutions.
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Example System:
- 2x + y - 3z = 5
- 3x - y + z = 4
- x + 2y + z = 7
Step-by-Step Solution (using Elimination):
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Step 1: Write down the system.
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Step 2: Choose two equations. Eliminate a variable. Here, work with equations (a) and (b) to eliminate 'y'.
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Step 3: Eliminating 'y' through addition:
- (2x + y - 3z) + (3x - y + z) = 5 + 4
- Resulting equation: 5x - 2z = 9 (Equation 4)
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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)
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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.
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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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