Linear Equations Solution Methods

Graphical Method

• A method to find the solution of a pair of linear equations in two variables by graphing them on a coordinate plane.
• The point of intersection of the two lines represents the solution of the system.
• If the lines are:
  • Parallel, the system has no solution.
  • Coincident, the system has infinitely many solutions.

Substitution Method

• A method to find the solution of a pair of linear equations in two variables by substituting one variable from one equation into the other equation.
• Steps:
  1. Solve one equation for one variable.
  2. Substitute the expression obtained in step 1 into the other equation.
  3. Solve the resulting equation to find the value of the variable.
  4. Substitute the value obtained in step 3 into one of the original equations to find the value of the other variable.

Elimination Method

• A method to find the solution of a pair of linear equations in two variables by making the coefficients of one variable opposite in both equations.
• Steps:
  1. Multiply both equations by necessary multiples such that the coefficients of one variable are the same.
  2. Add or subtract the equations to eliminate one variable.
  3. Solve the resulting equation to find the value of the variable.
  4. Substitute the value obtained in step 3 into one of the original equations to find the value of the other variable.

Consistent and Inconsistent Equations

• Consistent Equations: A system of linear equations that has at least one solution.
• Inconsistent Equations: A system of linear equations that has no solution.
• A system of linear equations is inconsistent if the lines are parallel and never intersect.

Dependent and Independent Equations

• Independent Equations: A system of linear equations where the equations are not multiples of each other.
• Dependent Equations: A system of linear equations where the equations are multiples of each other.
• A system of linear equations is dependent if the lines are coincident and have infinitely many solutions.

Graphical Method

• Graphing a pair of linear equations in two variables on a coordinate plane to find the solution
• The point of intersection of the two lines represents the solution of the system
• Parallel lines mean no solution, while coincident lines mean infinitely many solutions

Substitution Method

• Solving a pair of linear equations in two variables by substituting one variable from one equation into the other
• Steps involve solving one equation for one variable, substituting it into the other equation, and solving for the variables

Elimination Method

• Solving a pair of linear equations in two variables by making the coefficients of one variable opposite in both equations
• Steps involve multiplying equations to make coefficients equal, adding or subtracting to eliminate one variable, and solving for the variables

Consistent and Inconsistent Equations

• Consistent equations have at least one solution, while inconsistent equations have no solution
• Inconsistent equations have parallel lines that never intersect

Dependent and Independent Equations

• Independent equations are not multiples of each other, while dependent equations are multiples of each other
• Dependent equations have coincident lines with infinitely many solutions