Precalculus Examples

Determine if Dependent, Independent, or Inconsistent
,
Step 1
Solve the system of equations.
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Step 1.1
Multiply each equation by the value that makes the coefficients of opposite.
Step 1.2
Simplify.
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Step 1.2.1
Simplify the left side.
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Step 1.2.1.1
Simplify .
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Step 1.2.1.1.1
Apply the distributive property.
Step 1.2.1.1.2
Multiply.
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Step 1.2.1.1.2.1
Multiply by .
Step 1.2.1.1.2.2
Multiply by .
Step 1.2.2
Simplify the right side.
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Step 1.2.2.1
Multiply by .
Step 1.3
Add the two equations together to eliminate from the system.
Step 1.4
Since , the equations intersect at an infinite number of points.
Infinite number of solutions
Step 1.5
Solve one of the equations for .
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Step 1.5.1
Add to both sides of the equation.
Step 1.5.2
Divide each term in by and simplify.
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Step 1.5.2.1
Divide each term in by .
Step 1.5.2.2
Simplify the left side.
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Step 1.5.2.2.1
Cancel the common factor of .
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Step 1.5.2.2.1.1
Cancel the common factor.
Step 1.5.2.2.1.2
Divide by .
Step 1.5.2.3
Simplify the right side.
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Step 1.5.2.3.1
Simplify each term.
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Step 1.5.2.3.1.1
Cancel the common factor of and .
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Step 1.5.2.3.1.1.1
Factor out of .
Step 1.5.2.3.1.1.2
Cancel the common factors.
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Step 1.5.2.3.1.1.2.1
Factor out of .
Step 1.5.2.3.1.1.2.2
Cancel the common factor.
Step 1.5.2.3.1.1.2.3
Rewrite the expression.
Step 1.5.2.3.1.2
Move the negative in front of the fraction.
Step 1.5.2.3.1.3
Cancel the common factor of and .
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Step 1.5.2.3.1.3.1
Factor out of .
Step 1.5.2.3.1.3.2
Cancel the common factors.
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Step 1.5.2.3.1.3.2.1
Factor out of .
Step 1.5.2.3.1.3.2.2
Cancel the common factor.
Step 1.5.2.3.1.3.2.3
Rewrite the expression.
Step 1.6
The solution is the set of ordered pairs that make true.
Step 2
Since the system is always true, the equations are equal and the graphs are the same line. Thus, the system is dependent.
Dependent
Step 3
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