Linear Algebra Examples

Solve by Substitution 9x+2y=30 , 3x-4y=24
,
Step 1
Solve for in .
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Step 1.1
Subtract from both sides of the equation.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Simplify each term.
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Step 1.2.3.1.1
Cancel the common factor of and .
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Step 1.2.3.1.1.1
Factor out of .
Step 1.2.3.1.1.2
Cancel the common factors.
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Step 1.2.3.1.1.2.1
Factor out of .
Step 1.2.3.1.1.2.2
Cancel the common factor.
Step 1.2.3.1.1.2.3
Rewrite the expression.
Step 1.2.3.1.2
Move the negative in front of the fraction.
Step 2
Replace all occurrences of with in each equation.
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Step 2.1
Replace all occurrences of in with .
Step 2.2
Simplify the left side.
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Step 2.2.1
Simplify .
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Step 2.2.1.1
Simplify each term.
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Step 2.2.1.1.1
Apply the distributive property.
Step 2.2.1.1.2
Cancel the common factor of .
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Step 2.2.1.1.2.1
Cancel the common factor.
Step 2.2.1.1.2.2
Rewrite the expression.
Step 2.2.1.1.3
Cancel the common factor of .
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Step 2.2.1.1.3.1
Move the leading negative in into the numerator.
Step 2.2.1.1.3.2
Factor out of .
Step 2.2.1.1.3.3
Cancel the common factor.
Step 2.2.1.1.3.4
Rewrite the expression.
Step 2.2.1.1.4
Move the negative in front of the fraction.
Step 2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.3
Simplify terms.
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Step 2.2.1.3.1
Combine and .
Step 2.2.1.3.2
Combine the numerators over the common denominator.
Step 2.2.1.4
Simplify each term.
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Step 2.2.1.4.1
Simplify the numerator.
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Step 2.2.1.4.1.1
Factor out of .
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Step 2.2.1.4.1.1.1
Factor out of .
Step 2.2.1.4.1.1.2
Factor out of .
Step 2.2.1.4.1.1.3
Factor out of .
Step 2.2.1.4.1.2
Multiply by .
Step 2.2.1.4.1.3
Subtract from .
Step 2.2.1.4.1.4
Multiply by .
Step 2.2.1.4.2
Move the negative in front of the fraction.
Step 3
Solve for in .
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Step 3.1
Move all terms not containing to the right side of the equation.
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Step 3.1.1
Subtract from both sides of the equation.
Step 3.1.2
Subtract from .
Step 3.2
Multiply both sides of the equation by .
Step 3.3
Simplify both sides of the equation.
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Step 3.3.1
Simplify the left side.
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Step 3.3.1.1
Simplify .
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Step 3.3.1.1.1
Cancel the common factor of .
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Step 3.3.1.1.1.1
Move the leading negative in into the numerator.
Step 3.3.1.1.1.2
Move the leading negative in into the numerator.
Step 3.3.1.1.1.3
Factor out of .
Step 3.3.1.1.1.4
Cancel the common factor.
Step 3.3.1.1.1.5
Rewrite the expression.
Step 3.3.1.1.2
Cancel the common factor of .
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Step 3.3.1.1.2.1
Factor out of .
Step 3.3.1.1.2.2
Cancel the common factor.
Step 3.3.1.1.2.3
Rewrite the expression.
Step 3.3.1.1.3
Multiply.
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Step 3.3.1.1.3.1
Multiply by .
Step 3.3.1.1.3.2
Multiply by .
Step 3.3.2
Simplify the right side.
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Step 3.3.2.1
Cancel the common factor of .
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Step 3.3.2.1.1
Move the leading negative in into the numerator.
Step 3.3.2.1.2
Cancel the common factor.
Step 3.3.2.1.3
Rewrite the expression.
Step 4
Replace all occurrences of with in each equation.
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Step 4.1
Replace all occurrences of in with .
Step 4.2
Simplify the right side.
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Step 4.2.1
Simplify .
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Step 4.2.1.1
Simplify each term.
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Step 4.2.1.1.1
Cancel the common factor of and .
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Step 4.2.1.1.1.1
Factor out of .
Step 4.2.1.1.1.2
Cancel the common factors.
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Step 4.2.1.1.1.2.1
Factor out of .
Step 4.2.1.1.1.2.2
Cancel the common factor.
Step 4.2.1.1.1.2.3
Rewrite the expression.
Step 4.2.1.1.2
Multiply by .
Step 4.2.1.1.3
Move the negative in front of the fraction.
Step 4.2.1.1.4
Multiply .
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Step 4.2.1.1.4.1
Multiply by .
Step 4.2.1.1.4.2
Multiply by .
Step 4.2.1.2
Combine fractions.
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Step 4.2.1.2.1
Combine the numerators over the common denominator.
Step 4.2.1.2.2
Simplify the expression.
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Step 4.2.1.2.2.1
Add and .
Step 4.2.1.2.2.2
Divide by .
Step 5
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 6
The result can be shown in multiple forms.
Point Form:
Equation Form:
Step 7