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Questions and Answers
What is a system of linear equations?
What is a system of linear equations?
In the graphical method, what do the intersection points of two linear equations represent?
In the graphical method, what do the intersection points of two linear equations represent?
What method involves solving one equation for one variable and substituting it into another equation?
What method involves solving one equation for one variable and substituting it into another equation?
In the example provided, what was the value of x when solved?
In the example provided, what was the value of x when solved?
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What does it mean if a system of linear equations has infinitely many solutions?
What does it mean if a system of linear equations has infinitely many solutions?
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Why is solving systems of linear equations important in various fields?
Why is solving systems of linear equations important in various fields?
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What is a key step in using the elimination method to solve a system of linear equations?
What is a key step in using the elimination method to solve a system of linear equations?
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In Cramer's Rule, what does it mean if the determinant is zero?
In Cramer's Rule, what does it mean if the determinant is zero?
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When using Cramer's Rule, which determinants need to be calculated to solve for a variable?
When using Cramer's Rule, which determinants need to be calculated to solve for a variable?
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What is the significance of a determinant in solving systems of linear equations?
What is the significance of a determinant in solving systems of linear equations?
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In the elimination method, what happens when you add two equations together?
In the elimination method, what happens when you add two equations together?
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Why is mastering different methods of solving linear equations important?
Why is mastering different methods of solving linear equations important?
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Study Notes
Solving Systems of Linear Equations
When you encounter a pair or more linear equations with the same variables, you're facing a system of linear equations. Solving such systems is essential in various fields, from finance and engineering to social sciences. Let's delve into this fundamental topic by exploring the methods and strategies used to solve systems of linear equations.
Graphical Method
For two linear equations in two variables, we can plot the equations' graphs to find their points of intersection, which represent the solutions. By analyzing the number of intersection points, we can determine whether the system has no solutions, one solution, or infinitely many solutions.
Substitution Method
This method involves solving one equation for one variable (usually by isolating it) and then substituting this expression into the second equation to find the value of the other variable.
Example:
Solve the following system of linear equations:
[ x + y = 2 ] [ 3x + y = 5 ]
- Solve the first equation for (y): [ y = 2 - x ]
- Substitute (y = 2 - x) into the second equation: [ 3x + (2 - x) = 5 ]
- Simplify and solve for (x): [ 2x + 2 - x = 5 ] [ 2x = 3 ] [ x = \frac{3}{2} ]
- Substitute (x = \frac{3}{2}) back into the equation (y = 2 - x): [ y = 2 - \frac{3}{2} = \frac{2}{2} - \frac{3}{2} = -\frac{1}{2} ]
So, the solution is (\left(\frac{3}{2}, -\frac{1}{2}\right)).
Elimination Method
This method involves manipulating the equations to eliminate one variable, either by adding or subtracting them.
Example:
Solve the following system of linear equations:
[ x + y = 2 ] [ -x + 2y = 6 ]
- Multiply the first equation by (-)1: [ -x - y = -2 ]
- Add the two equations: [ (-x - y) + (-x + 2y) = -2 + 6 ]
- Simplify: [ 0 + y = 4 ]
- Solve for (y): [ y = 4 ]
- Substitute (y = 4) into either of the original equations to find (x): [ x + 4 = 2 ] [ x = -2 ]
So, the solution is ((-2, 4)).
Cramer's Rule
This method is an extension of the elimination method and is particularly useful for solving systems of three linear equations in three variables.
Example:
Solve the following system of linear equations using Cramer's Rule:
[ x + 2y - z = 5 ] [ 3x - y + 2z = 4 ] [ 2x + y - z = 1 ]
-
Form the determinants for (x): [ \begin{vmatrix} 1 & 2 & -1 \ 3 & -1 & 2 \ 2 & 1 & -1 \end{vmatrix} ]
-
Form the determinants for the other variables: [ \begin{vmatrix} 5 & 2 & -1 \ 4 & -1 & 2 \ 1 & 1 & -1 \end{vmatrix} \quad \begin{vmatrix} 1 & 5 & -1 \ 3 & 4 & 2 \ 2 & 1 & -1 \end{vmatrix} ]
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Calculate the determinants and solve for (x): [ x = \frac{\begin{vmatrix} 5 & 2 & -1 \ 4 & -1 & 2 \ 1 & 1 & -1 \end{vmatrix}}{\begin{vmatrix} 1 & 2 & -1 \ 3 & -1 & 2 \ 2 & 1 & -1 \end{vmatrix}} ]
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Simplify the determinants: [ x = \frac{-3}{-3} = 1 ]
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Find (y) and (z) by replacing (x) with 1 in the original equations.
In this case, the system has infinitely many solutions due to the determinant being zero for (z).
Final Thoughts
As we've seen, solving systems of linear equations can be achieved through multiple methods. Each method has its strengths and weaknesses, and each might be more suitable for a specific system depending on its structure. Mastering these methods will open doors to a world of problem-solving in various fields.
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Description
Explore methods like graphical, substitution, elimination, and Cramer's Rule for solving systems of linear equations. Learn how to find solutions using determinants, manipulation of equations, and variable isolation. Mastering these methods is crucial for problem-solving in fields like finance, engineering, and social sciences.
