Solving Systems of Linear Equations

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.