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Questions and Answers
What is the equivalent of the second equation in Example 1 if we multiply it by 4?
What is the equivalent of the second equation in Example 1 if we multiply it by 4?
What is the value of y in the system of linear equations: -7x - 4y = 9 and x + 2y = 3?
What is the value of y in the system of linear equations: -7x - 4y = 9 and x + 2y = 3?
What operation is performed on the two equations in Example 1 to get 25x = -150?
What operation is performed on the two equations in Example 1 to get 25x = -150?
What is the solution to the system of linear equations: -3x - 9y = 66 and -7x + 4y = -71?
What is the solution to the system of linear equations: -3x - 9y = 66 and -7x + 4y = -71?
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What is the purpose of multiplying both equations by a certain number in the system of linear equations?
What is the purpose of multiplying both equations by a certain number in the system of linear equations?
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In Example 1, we get x = -6 by subtracting the two original equations.
In Example 1, we get x = -6 by subtracting the two original equations.
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The solution to the system of linear equations in Example 2 is x = 3 and y = -3.
The solution to the system of linear equations in Example 2 is x = 3 and y = -3.
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In Example 3, we multiply the first equation by 9 and the second equation by 4 to eliminate the y variable.
In Example 3, we multiply the first equation by 9 and the second equation by 4 to eliminate the y variable.
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The system of linear equations in Example 1 has no solution.
The system of linear equations in Example 1 has no solution.
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In Example 2, we solve for x before solving for y.
In Example 2, we solve for x before solving for y.
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Study Notes
Solving Systems of Linear Equations
- A system of linear equations consists of two or more equations with two or more unknowns.
- To solve a system of linear equations, we need to find the values of the unknowns that satisfy both equations.
Graphical Representation
- The graphical representation of a system of linear equations is a pair of lines that intersect at a single point.
- The point of intersection represents the solution to the system of linear equations.
Elimination Method
- The elimination method is used to solve systems of linear equations by eliminating one of the unknowns.
- To eliminate one unknown, we can multiply one or both of the equations by a constant to make the coefficients of the unknown to be eliminated equal.
- Once the unknown is eliminated, we can solve for the other unknown.
Example 1: Simple System of Linear Equations
- The system of linear equations: 9x - 4y = -78 and 4x + y = -18
- By multiplying the second equation by 4, we get 16x + 4y = -72
- Adding the two equations, we get 25x = -150
- Solving for x, we get x = -6
- Substituting x into one of the original equations, we get y = 6
- The solution to the system of linear equations is x = -6 and y = 6
Example 2: Another System of Linear Equations
- The system of linear equations: -7x - 4y = 9 and x + 2y = 3
- By multiplying the second equation by 7, we get 7x + 14y = 21
- Adding the two equations, we get 10y = 30
- Solving for y, we get y = 3
- Substituting y into one of the original equations, we get x = -3
- The solution to the system of linear equations is x = -3 and y = 3
Example 3: More Complex System of Linear Equations
- The system of linear equations: -3x - 9y = 66 and -7x + 4y = -71
- By multiplying the first equation by 4 and the second equation by 9, we get -20x - 36y = 264 and -63x + 36y = -639
- Adding the two equations, we get -83x = -375
- Solving for x, we get x = 5
- Substituting x into one of the original equations, we get y = -9
- The solution to the system of linear equations is x = 5 and y = -9
Solving Systems of Linear Equations
- A system of linear equations consists of two or more equations with two or more unknowns.
- The goal is to find the values of the unknowns that satisfy all equations.
Graphical Representation
- Graphical representation shows a pair of lines that intersect at a single point.
- The point of intersection represents the solution to the system of linear equations.
Elimination Method
- Elimination method is used to solve systems of linear equations by eliminating one unknown.
- To eliminate one unknown, multiply one or both equations by a constant to make coefficients equal.
- Once the unknown is eliminated, solve for the other unknown.
Example 1: Simple System of Linear Equations
- System: 9x - 4y = -78 and 4x + y = -18.
- Multiply second equation by 4 to get 16x + 4y = -72.
- Add equations to get 25x = -150.
- Solve for x to get x = -6.
- Substitute x into an original equation to get y = 6.
- Solution: x = -6 and y = 6.
Example 2: Another System of Linear Equations
- System: -7x - 4y = 9 and x + 2y = 3.
- Multiply second equation by 7 to get 7x + 14y = 21.
- Add equations to get 10y = 30.
- Solve for y to get y = 3.
- Substitute y into an original equation to get x = -3.
- Solution: x = -3 and y = 3.
Example 3: More Complex System of Linear Equations
- System: -3x - 9y = 66 and -7x + 4y = -71.
- Multiply first equation by 4 and second equation by 9 to get -20x - 36y = 264 and -63x + 36y = -639.
- Add equations to get -83x = -375.
- Solve for x to get x = 5.
- Substitute x into an original equation to get y = -9.
- Solution: x = 5 and y = -9.
Solving Systems of Linear Equations
- A system of linear equations consists of two or more equations with two or more unknowns.
- The goal is to find the values of the unknowns that satisfy all equations.
Graphical Representation
- Graphical representation shows a pair of lines that intersect at a single point.
- The point of intersection represents the solution to the system of linear equations.
Elimination Method
- Elimination method is used to solve systems of linear equations by eliminating one unknown.
- To eliminate one unknown, multiply one or both equations by a constant to make coefficients equal.
- Once the unknown is eliminated, solve for the other unknown.
Example 1: Simple System of Linear Equations
- System: 9x - 4y = -78 and 4x + y = -18.
- Multiply second equation by 4 to get 16x + 4y = -72.
- Add equations to get 25x = -150.
- Solve for x to get x = -6.
- Substitute x into an original equation to get y = 6.
- Solution: x = -6 and y = 6.
Example 2: Another System of Linear Equations
- System: -7x - 4y = 9 and x + 2y = 3.
- Multiply second equation by 7 to get 7x + 14y = 21.
- Add equations to get 10y = 30.
- Solve for y to get y = 3.
- Substitute y into an original equation to get x = -3.
- Solution: x = -3 and y = 3.
Example 3: More Complex System of Linear Equations
- System: -3x - 9y = 66 and -7x + 4y = -71.
- Multiply first equation by 4 and second equation by 9 to get -20x - 36y = 264 and -63x + 36y = -639.
- Add equations to get -83x = -375.
- Solve for x to get x = 5.
- Substitute x into an original equation to get y = -9.
- Solution: x = 5 and y = -9.
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Description
Learn how to solve systems of linear equations through graphical representation and elimination methods. Find the values of unknowns that satisfy both equations.
