math 71a 3.1 – systems of linear equations in two variables 1
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Math 71A
3.1 – Systems of Linear Equations in Two Variables
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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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Ex 4.Solve by the addition method:
Addition Method
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Ex 5.Solve by the addition method (Hint: clear the fractions first!):
Addition Method
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No Solution or Infinitely Many Solutions
Exactly one solution No solution Infinitely many solutions
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No Solution or Infinitely Many Solutions
Exactly one solution No solution Infinitely many solutions
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No Solution or Infinitely Many Solutions
Exactly one solution No solution Infinitely many solutions
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No Solution or Infinitely Many Solutions
Exactly one solution No solution Infinitely many solutions
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No Solution or Infinitely Many Solutions
Exactly one solution No solution Infinitely many solutions
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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