Finite Math Examples

Find the Slope for Each Equation 2x+5y=10 , x-3y=3
,
Step 1
Rewrite in slope-intercept form.
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Step 1.1
The slope-intercept form is , where is the slope and is the y-intercept.
Step 1.2
Subtract from both sides of the equation.
Step 1.3
Divide each term in by and simplify.
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Step 1.3.1
Divide each term in by .
Step 1.3.2
Simplify the left side.
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Step 1.3.2.1
Cancel the common factor of .
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Step 1.3.2.1.1
Cancel the common factor.
Step 1.3.2.1.2
Divide by .
Step 1.3.3
Simplify the right side.
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Step 1.3.3.1
Simplify each term.
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Step 1.3.3.1.1
Divide by .
Step 1.3.3.1.2
Move the negative in front of the fraction.
Step 1.4
Write in form.
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Step 1.4.1
Reorder and .
Step 1.4.2
Reorder terms.
Step 1.4.3
Remove parentheses.
Step 2
Using the slope-intercept form, the slope is .
Step 3
Rewrite in slope-intercept form.
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Step 3.1
The slope-intercept form is , where is the slope and is the y-intercept.
Step 3.2
Subtract from both sides of the equation.
Step 3.3
Divide each term in by and simplify.
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Step 3.3.1
Divide each term in by .
Step 3.3.2
Simplify the left 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
Cancel the common factor.
Step 3.3.2.1.2
Divide by .
Step 3.3.3
Simplify the right side.
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Step 3.3.3.1
Simplify each term.
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Step 3.3.3.1.1
Divide by .
Step 3.3.3.1.2
Dividing two negative values results in a positive value.
Step 3.4
Write in form.
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Step 3.4.1
Reorder and .
Step 3.4.2
Reorder terms.
Step 4
Using the slope-intercept form, the slope is .
Step 5
Set up the system of equations to find any points of intersection.
Step 6
Solve the system of equations to find the point of intersection.
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Step 6.1
Add to both sides of the equation.
Step 6.2
Replace all occurrences of with in each equation.
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Step 6.2.1
Replace all occurrences of in with .
Step 6.2.2
Simplify the left side.
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Step 6.2.2.1
Simplify .
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Step 6.2.2.1.1
Simplify each term.
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Step 6.2.2.1.1.1
Apply the distributive property.
Step 6.2.2.1.1.2
Multiply by .
Step 6.2.2.1.1.3
Multiply by .
Step 6.2.2.1.2
Add and .
Step 6.3
Solve for in .
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Step 6.3.1
Move all terms not containing to the right side of the equation.
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Step 6.3.1.1
Subtract from both sides of the equation.
Step 6.3.1.2
Subtract from .
Step 6.3.2
Divide each term in by and simplify.
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Step 6.3.2.1
Divide each term in by .
Step 6.3.2.2
Simplify the left side.
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Step 6.3.2.2.1
Cancel the common factor of .
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Step 6.3.2.2.1.1
Cancel the common factor.
Step 6.3.2.2.1.2
Divide by .
Step 6.4
Replace all occurrences of with in each equation.
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Step 6.4.1
Replace all occurrences of in with .
Step 6.4.2
Simplify the right side.
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Step 6.4.2.1
Simplify .
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Step 6.4.2.1.1
Multiply .
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Step 6.4.2.1.1.1
Combine and .
Step 6.4.2.1.1.2
Multiply by .
Step 6.4.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 6.4.2.1.3
Combine and .
Step 6.4.2.1.4
Combine the numerators over the common denominator.
Step 6.4.2.1.5
Simplify the numerator.
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Step 6.4.2.1.5.1
Multiply by .
Step 6.4.2.1.5.2
Add and .
Step 6.5
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 7
Since the slopes are different, the lines will have exactly one intersection point.
Step 8