Module 11

Module 11: Solving Systems of Linear Equations

Lesson 11.1: Solving Linear Systems by Graphing

• A system of linear equations (linear system) consists of two or more linear equations that have the same variables

• A solution of a system of linear equations with two variables is any ordered pair that satisfies all of the equations in the system.

Multiple Choice

any ordered pair that satisfies all of the equations in the system

• system of linear equations

• solution of a system of linear equations

Multiple Choice

a system of equations in which all the equations are linear

• system of linear equations

• solution of a system of linear equations

Types of Systems of Linear Equations

• A consistent system is a system with at least one solution. Consistent systems can be independent or dependent.

• An independent system has exactly one solution. The graph of an independent system consists of two lines that intersect at exactly one point.

• A dependent system has infinitely many solutions. The graph of a dependent system consists of two coincident lines, or the same line.

• A system that has no solution is an inconsistent system. An inconsistent system consists of two parallel lines.

Multiple Choice

a system with at least one solution

• consistent system

• inconsistent system

• independent system

• dependent system

Multiple Choice

a system that has no solution

• consistent system

• inconsistent system

• independent system

• dependent system

Multiple Choice

a system that has exactly one solution

• consistent system

• inconsistent system

• independent system

• dependent system

Multiple Choice

a system that has infinitely many solutions

• consistent system

• inconsistent system

• independent system

• dependent system

Solve the system of linear equations by graphing.

• 2x + y = 6

• -x + y = 3

• Graph the two equations and see where they intersect.

Independent consistent equations intersect at a single point. The point where they intersect is the solution to the system.

An inconsistent system has no solutions. The lines are parallel and do not intersect.

A dependent system has infinitely many solutions. The graphs are the same line, simply written in different forms.

Multiple Select

Select all systems that are consistent.

Multiple Select

Select all systems that are inconsistent.

Multiple Select

Select all systems that are independent.

Multiple Select

Select all systems that are dependent.

Lesson 11.2: Solving Linear Systems by Substitution

Solving Consistent, Independent Linear Systems by Substitution

• The substitution method is used to solve a system of equations by solving an equation for one variable and substituting the resulting expression into the other equation.

• Step 1: Solve one of the equations for one of its variables.

• Step 2: Substitute the expression from Step 1 into the other equation and solve for the other variable.

• Step 3: Substitute the value from Step 2 into either original equation and solve to find the value of the other variable.

Special Cases

• You can use the substitution method for systems of linear equations that have infinitely many solutions and for systems that have no solutions.

• Step 1: Solve one of the equations for one of its variables.

• Step 2: Substitute the expression from Step 1 into the other equation and solve for the other variable.

• If you get a resulting equation that is false, the system has no solutions. ex: 4 = 6

• If you get a resulting equation that is true, the system has infinitely many solutions. ex: 2 = 2

Lesson 11.3: Solving Linear Systems by Adding or Subtracting

Open Ended

What are the two methods of solving linear equations that we've learned about so far?

Solving Linear Systems by Adding or Subtracting

• The elimination method is used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations in the system.

• Step 1: Add or subtract the equations to eliminate one variable, and then solve for the other variable.

• Step 2: Substitute the value into either original equation to find the value of the eliminated variable.

• Step 3: Write the solution as an ordered pair.

Special Cases

• You can use the elimination method for systems of linear equations that have infinitely many solutions and for systems that have no solutions.

• Add or subtract the equations to eliminate one variable, and then solve for the other variable.

• If you get a resulting equation that is false, the system has no solutions. ex: 4 = 6

• If you get a resulting equation that is true, the system has infinitely many solutions. ex: 2 = 2

• 
  

Lesson 11.4: Solving Linear Systems by Multiplying First

Solving Linear Systems by Multiplying First

• If you can't directly add or subtract equations to use elimination, you can multiply one or both of the equations by a constant so that elimination is possible.

• Step 1: Decide which variable to eliminate.

• Step 2: Multiply one or both equations by a constant so that adding or subtracting the equations will eliminate the variable.

• Step 3: Solve the system using the elimination method.

Multiple Choice

a method used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations of the system

• substitution method

• graphing method

• elimination method

Multiple Choice

a method used to solve systems of equations by graphing the equations and looking for points of intersection

• substitution method

• graphing method

• elimination method

Multiple Choice

a method used to solve systems of equations by solving an equation for one variable and substituting the resulting expression into the other equation

• substitution method

• graphing method

• elimination method