Solving Systems of Linear Equations

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Questions and Answers

What is the equivalent of the second equation in Example 1 if we multiply it by 4?

  • 16x - 4y = 18
  • 16x + 4y = 72
  • 16x + 4y = -72 (correct)
  • 16x - 4y = -72

What is the value of y in the system of linear equations: -7x - 4y = 9 and x + 2y = 3?

  • y = 4
  • y = 2
  • y = 3 (correct)
  • y = 5

What operation is performed on the two equations in Example 1 to get 25x = -150?

  • Division
  • Multiplication
  • Subtraction
  • Addition (correct)

What is the solution to the system of linear equations: -3x - 9y = 66 and -7x + 4y = -71?

<p>x = 5, y = -9 (A)</p> Signup and view all the answers

What is the purpose of multiplying both equations by a certain number in the system of linear equations?

<p>To make the coefficients of x or y opposites (C)</p> Signup and view all the answers

In Example 1, we get x = -6 by subtracting the two original equations.

<p>False (B)</p> Signup and view all the answers

The solution to the system of linear equations in Example 2 is x = 3 and y = -3.

<p>False (B)</p> Signup and view all the answers

In Example 3, we multiply the first equation by 9 and the second equation by 4 to eliminate the y variable.

<p>False (B)</p> Signup and view all the answers

The system of linear equations in Example 1 has no solution.

<p>False (B)</p> Signup and view all the answers

In Example 2, we solve for x before solving for y.

<p>False (B)</p> Signup and view all the answers

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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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