Solving Systems of Linear Equations

Solving Systems Of Equations

• A system of linear equations is a set of two or more linear equations that must be true at the same time.
• Systems of equations can be solved using:
  • Substitution method
    • Elimination method
  • Graphical method

Substitution Method

• Solve one equation for one variable
• Substitute the expression into the other equation
• Solve for the other variable
• Substitute the value of the variable back into one of the original equations to find the value of the other variable

Elimination Method

• Make the coefficients of one variable opposites
• Add the equations to eliminate the variable
• Solve for the remaining variable
• Substitute the value of the variable back into one of the original equations to find the value of the other variable

Slope-Intercept Form

• The slope-intercept form of a linear equation is given by:
  • y = mx + b
  • m is the slope (rate of change)
  • b is the y-intercept (point where the line crosses the y-axis)
• Slope-intercept form is useful for:
  • Graphing linear equations
  • Identifying the slope and y-intercept of a line

Standard Form

• The standard form of a linear equation is given by:
  • Ax + By = C
  • A, B, and C are integers
  • A and B are not both zero
• Standard form is useful for:
  • Writing linear equations in a consistent format
  • Solving systems of linear equations
• Standard form can be converted to slope-intercept form by:
  • Rearranging the equation to isolate y
  • Dividing by the coefficient of y (if necessary)

Systems of Linear Equations

• A system consists of two or more linear equations that must be true at the same time
• Can be solved using three methods: Substitution, Elimination, and Graphical method

Solving Systems of Equations using Substitution Method

• Solve one equation for one variable
• Substitute the expression into the other equation to solve for the other variable
• Substitute the value of the variable back into one of the original equations to find the value of the other variable

Solving Systems of Equations using Elimination Method

• Make the coefficients of one variable opposites
• Add the equations to eliminate the variable
• Solve for the remaining variable
• Substitute the value of the variable back into one of the original equations to find the value of the other variable

Slope-Intercept Form

• Equation is given by: y = mx + b
• m represents the slope (rate of change)
• b represents the y-intercept (point where the line crosses the y-axis)
• Useful for graphing linear equations and identifying the slope and y-intercept of a line

Standard Form

• Equation is given by: Ax + By = C
• A, B, and C are integers
• A and B are not both zero
• Useful for writing linear equations in a consistent format and solving systems of linear equations
• Can be converted to slope-intercept form by rearranging the equation to isolate y and dividing by the coefficient of y (if necessary)