Solving Systems of Equations

Solving Systems of Equations

• A system of equations is a set of two or more equations with the same variables.
• The solution to a system of equations is a set of values for the variables that satisfy all the equations in the system.
• Systems of equations can be solved using various methods, including graphing, substitution, and elimination.

Graphing Method

• Graph each equation in the system on the same coordinate plane.
• The point where the graphs intersect represents the solution to the system.
• This method is best suited for systems with linear equations that are easily graphed.
• If the lines do not intersect, the system has no solution (inconsistent). If the lines are the same, the system has infinitely many solutions (dependent).

Substitution Method

• Solve one equation for one variable.
• Substitute the expression for that variable into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute the value of the solved variable back into the original equation to find the value of the other variable.
• This method is useful when one or more of the equations can be easily solved for a variable.

Elimination Method

• Adjust equations (if necessary) by multiplying one or both equations by a constant to have opposite coefficients for one variable.
• Add or subtract the equations to eliminate one variable.
• Solve the resulting equation for the remaining variable.
• Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.
• This is often a more efficient method when both equations are in standard form.

Types of Systems

• Consistent and Independent: The system has one unique solution, corresponding to a single point of intersection.
• Consistent and Dependent: The system has infinitely many solutions, corresponding to the same lines.
• Inconsistent: The system has no solution, corresponding to parallel lines.

Examples

• Example 1 (Substitution):

  • Equation 1: y = 2x + 1
  • Equation 2: y = -x + 4
  • Substituting the first equation into the second gives: 2x+1 = -x+4
  • Solving for x yields the value of x = 1.
  • Substitute x = 1 into either equation to solve for y. Using the first equation, y = 2(1) + 1 = 3.
  • The solution is (1,3).

• Example 2 (Elimination):

  • Equation 1: 2x + 3y = 7
  • Equation 2: x – 3y = 2
  • Adding the equations results in 3x = 9, which solves to x = 3.
  • Substitute x = 3 into either original equation to solve for y. Using the second equation, 3 – 3y = 2, gives 3y = 1, and y = 1/3.
  • The solution is (3, 1/3).

Applications

• Solving systems of equations has applications in various fields, including:
  • Physics, describing motion or equilibrium.
  • Engineering, in designing structures or determining optimal designs.
  • Business, optimizing production and pricing strategies.
  • Economics, modelling supply and demand or the interaction of markets.

Important Considerations

• Consider the type of system and the form of the equations.
• Select the appropriate method for solving, considering efficiency and ease of calculation.
• Check the solution by substituting the values of the variables into the original equations.
• Be aware that some systems may have no solution or infinitely many solutions.