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Math 51

4.1/4.2/4.3 – Solving Systems of Linear Equations

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Suppose we have two linear equations:

Together, they make a ________________________________ (also called a ___________________).

Solutions must satisfy all equations.

For example, is a solution to the above system since it satisfies both equations.

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Suppose we have two linear equations:

Together, they make a ________________________________ (also called a ___________________).

Solutions must satisfy all equations.

For example, is a solution to the above system since it satisfies both equations.

system of linear equations

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Suppose we have two linear equations:

Together, they make a ________________________________ (also called a ___________________).

Solutions must satisfy all equations.

For example, is a solution to the above system since it satisfies both equations.

system of linear equationslinear system

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Suppose we have two linear equations:

Together, they make a ________________________________ (also called a ___________________).

Solutions must satisfy all equations.

For example, is a solution to the above system since it satisfies both equations.

system of linear equationslinear system

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Suppose we have two linear equations:

Together, they make a ________________________________ (also called a ___________________).

Solutions must satisfy all equations.

For example, is a solution to the above system since it satisfies both equations.

system of linear equationslinear system

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Ex 1.Determine if is a solution to the following system:

Is a solution?

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Besides , are there any other solutions to the following system?

We can show this by graphing the two lines.

Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

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Graphing Systems of Linear Equations

is the onlysolution to system

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Ex 2.Solve by the substitution method:

Substitution Method

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1. Solve one equation for or 2. Substitute the expression we get for or into

the other equation3. This makes an equation with one variable

that we can solve4. After that, we can find the other variable

Substitution Method

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1. Solve one equation for or 2. Substitute the expression we get for or into

the other equation3. This makes an equation with one variable

that we can solve4. After that, we can find the other variable

Substitution Method

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1. Solve one equation for or 2. Substitute the expression we get for or into

the other equation3. This makes an equation with one variable

that we can solve4. After that, we can find the other variable

Substitution Method

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1. Solve one equation for or 2. Substitute the expression we get for or into

the other equation3. This makes an equation with one variable

that we can solve4. After that, we can find the other variable

Substitution Method

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Ex 3.Solve by the substitution method:

Substitution Method

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The elimination method is just another way to solve linear systems.

Ex 4.Solve by the elimination method:

Elimination Method

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How to solve a system using the elimination method:1. Write equations in form 2. Use multiplication to make coefficients of or add up to zero3. Add equations (after which one variable should be eliminated)4. This makes an equation with one variable that we can solve5. After that, we can find the other variable

Elimination Method

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How to solve a system using the elimination method:1. Write equations in form 2. Use multiplication to make coefficients of or add up to zero3. Add equations (after which one variable should be eliminated)4. This makes an equation with one variable that we can solve5. After that, we can find the other variable

Elimination Method

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How to solve a system using the elimination method:1. Write equations in form 2. Use multiplication to make coefficients of or add up to zero3. Add equations (after which one variable should be eliminated)4. This makes an equation with one variable that we can solve5. After that, we can find the other variable

Elimination Method

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How to solve a system using the elimination method:1. Write equations in form 2. Use multiplication to make coefficients of or add up to zero3. Add equations (after which one variable should be eliminated)4. This makes an equation with one variable that we can solve5. After that, we can find the other variable

Elimination Method

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How to solve a system using the elimination method:1. Write equations in form 2. Use multiplication to make coefficients of or add up to zero3. Add equations (after which one variable should be eliminated)4. This makes an equation with one variable that we can solve5. After that, we can find the other variable

Elimination Method

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Ex 5.Solve by the elimination method (Hint: clear the fractions first!):

Elimination Method

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No Solution or Infinitely Many Solutions

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No Solution or Infinitely Many Solutions

intersectonce

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No Solution or Infinitely Many Solutions

Exactly one solution

intersectonce

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No Solution or Infinitely Many Solutions

Exactly one solution

intersectonce

parallel

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No Solution or Infinitely Many Solutions

Exactly one solution No solution

intersectonce

parallel

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No Solution or Infinitely Many Solutions

Exactly one solution No solution

intersectonce

parallel

identical

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No Solution or Infinitely Many Solutions

Exactly one solution No solution Infinitely many solutions

intersectonce

parallel

identical

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Ex 6.Solve the system:

No Solution or Infinitely Many Solutions

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Ex 7.Solve the system:

No Solution or Infinitely Many Solutions

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Summary

When solving a system of linear equations, if you…

…get a false statement, then the system is inconsistent, the solution set is , and the lines are parallel.

…get a true statement, then the system is dependent, the solution set is , and the lines are identical.