Finite Math Examples

Find the Slope for Each Equation 7/4x-5/2y=2 , 1/4x+7/2y=2
,
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
Simplify the left side.
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Step 1.2.1
Simplify each term.
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Step 1.2.1.1
Combine and .
Step 1.2.1.2
Combine and .
Step 1.2.1.3
Move to the left of .
Step 1.3
Subtract from both sides of the equation.
Step 1.4
Multiply both sides of the equation by .
Step 1.5
Simplify both sides of the equation.
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Step 1.5.1
Simplify the left side.
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Step 1.5.1.1
Simplify .
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Step 1.5.1.1.1
Cancel the common factor of .
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Step 1.5.1.1.1.1
Move the leading negative in into the numerator.
Step 1.5.1.1.1.2
Move the leading negative in into the numerator.
Step 1.5.1.1.1.3
Factor out of .
Step 1.5.1.1.1.4
Cancel the common factor.
Step 1.5.1.1.1.5
Rewrite the expression.
Step 1.5.1.1.2
Cancel the common factor of .
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Step 1.5.1.1.2.1
Factor out of .
Step 1.5.1.1.2.2
Cancel the common factor.
Step 1.5.1.1.2.3
Rewrite the expression.
Step 1.5.1.1.3
Multiply.
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Step 1.5.1.1.3.1
Multiply by .
Step 1.5.1.1.3.2
Multiply by .
Step 1.5.2
Simplify the right side.
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Step 1.5.2.1
Simplify .
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Step 1.5.2.1.1
Apply the distributive property.
Step 1.5.2.1.2
Multiply .
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Step 1.5.2.1.2.1
Multiply by .
Step 1.5.2.1.2.2
Combine and .
Step 1.5.2.1.2.3
Multiply by .
Step 1.5.2.1.3
Cancel the common factor of .
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Step 1.5.2.1.3.1
Move the leading negative in into the numerator.
Step 1.5.2.1.3.2
Move the leading negative in into the numerator.
Step 1.5.2.1.3.3
Factor out of .
Step 1.5.2.1.3.4
Factor out of .
Step 1.5.2.1.3.5
Cancel the common factor.
Step 1.5.2.1.3.6
Rewrite the expression.
Step 1.5.2.1.4
Multiply by .
Step 1.5.2.1.5
Simplify the expression.
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Step 1.5.2.1.5.1
Multiply by .
Step 1.5.2.1.5.2
Multiply by .
Step 1.5.2.1.5.3
Move the negative in front of the fraction.
Step 1.6
Reorder and .
Step 1.7
Reorder terms.
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
Simplify the left side.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Combine and .
Step 3.2.1.2
Combine and .
Step 3.3
Subtract from both sides of the equation.
Step 3.4
Multiply both sides of the equation by .
Step 3.5
Simplify both sides of the equation.
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Step 3.5.1
Simplify the left side.
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Step 3.5.1.1
Simplify .
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Step 3.5.1.1.1
Cancel the common factor of .
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Step 3.5.1.1.1.1
Cancel the common factor.
Step 3.5.1.1.1.2
Rewrite the expression.
Step 3.5.1.1.2
Cancel the common factor of .
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Step 3.5.1.1.2.1
Factor out of .
Step 3.5.1.1.2.2
Cancel the common factor.
Step 3.5.1.1.2.3
Rewrite the expression.
Step 3.5.2
Simplify the right side.
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Step 3.5.2.1
Simplify .
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Step 3.5.2.1.1
Apply the distributive property.
Step 3.5.2.1.2
Multiply .
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Step 3.5.2.1.2.1
Combine and .
Step 3.5.2.1.2.2
Multiply by .
Step 3.5.2.1.3
Cancel the common factor of .
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Step 3.5.2.1.3.1
Move the leading negative in into the numerator.
Step 3.5.2.1.3.2
Factor out of .
Step 3.5.2.1.3.3
Cancel the common factor.
Step 3.5.2.1.3.4
Rewrite the expression.
Step 3.5.2.1.4
Multiply by .
Step 3.5.2.1.5
Simplify the expression.
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Step 3.5.2.1.5.1
Multiply by .
Step 3.5.2.1.5.2
Move the negative in front of the fraction.
Step 3.6
Reorder and .
Step 3.7
Write in form.
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Step 3.7.1
Reorder terms.
Step 3.7.2
Remove parentheses.
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
Solve for in .
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Step 6.1.1
Simplify each term.
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Step 6.1.1.1
Combine and .
Step 6.1.1.2
Combine and .
Step 6.1.1.3
Move to the left of .
Step 6.1.2
Add to both sides of the equation.
Step 6.1.3
Multiply both sides of the equation by .
Step 6.1.4
Simplify both sides of the equation.
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Step 6.1.4.1
Simplify the left side.
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Step 6.1.4.1.1
Simplify .
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Step 6.1.4.1.1.1
Cancel the common factor of .
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Step 6.1.4.1.1.1.1
Cancel the common factor.
Step 6.1.4.1.1.1.2
Rewrite the expression.
Step 6.1.4.1.1.2
Cancel the common factor of .
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Step 6.1.4.1.1.2.1
Factor out of .
Step 6.1.4.1.1.2.2
Cancel the common factor.
Step 6.1.4.1.1.2.3
Rewrite the expression.
Step 6.1.4.2
Simplify the right side.
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Step 6.1.4.2.1
Simplify .
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Step 6.1.4.2.1.1
Apply the distributive property.
Step 6.1.4.2.1.2
Multiply .
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Step 6.1.4.2.1.2.1
Combine and .
Step 6.1.4.2.1.2.2
Multiply by .
Step 6.1.4.2.1.3
Cancel the common factor of .
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Step 6.1.4.2.1.3.1
Factor out of .
Step 6.1.4.2.1.3.2
Cancel the common factor.
Step 6.1.4.2.1.3.3
Rewrite the expression.
Step 6.1.4.2.1.4
Combine and .
Step 6.1.4.2.1.5
Multiply by .
Step 6.1.4.2.1.6
Combine and .
Step 6.1.5
Reorder and .
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
Cancel the common factor of .
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Step 6.2.2.1.1.2.1
Factor out of .
Step 6.2.2.1.1.2.2
Factor out of .
Step 6.2.2.1.1.2.3
Cancel the common factor.
Step 6.2.2.1.1.2.4
Rewrite the expression.
Step 6.2.2.1.1.3
Multiply by .
Step 6.2.2.1.1.4
Multiply by .
Step 6.2.2.1.1.5
Cancel the common factor of .
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Step 6.2.2.1.1.5.1
Factor out of .
Step 6.2.2.1.1.5.2
Cancel the common factor.
Step 6.2.2.1.1.5.3
Rewrite the expression.
Step 6.2.2.1.1.6
Combine and .
Step 6.2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 6.2.2.1.3.1
Multiply by .
Step 6.2.2.1.3.2
Multiply by .
Step 6.2.2.1.4
Combine the numerators over the common denominator.
Step 6.2.2.1.5
Simplify each term.
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Step 6.2.2.1.5.1
Simplify the numerator.
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Step 6.2.2.1.5.1.1
Factor out of .
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Step 6.2.2.1.5.1.1.1
Factor out of .
Step 6.2.2.1.5.1.1.2
Factor out of .
Step 6.2.2.1.5.1.1.3
Factor out of .
Step 6.2.2.1.5.1.2
Multiply by .
Step 6.2.2.1.5.1.3
Add and .
Step 6.2.2.1.5.2
Cancel the common factor of and .
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Step 6.2.2.1.5.2.1
Factor out of .
Step 6.2.2.1.5.2.2
Cancel the common factors.
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Step 6.2.2.1.5.2.2.1
Factor out of .
Step 6.2.2.1.5.2.2.2
Cancel the common factor.
Step 6.2.2.1.5.2.2.3
Rewrite the expression.
Step 6.2.2.1.5.3
Move to the left of .
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
To write as a fraction with a common denominator, multiply by .
Step 6.3.1.3
Combine and .
Step 6.3.1.4
Combine the numerators over the common denominator.
Step 6.3.1.5
Simplify the numerator.
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Step 6.3.1.5.1
Multiply by .
Step 6.3.1.5.2
Subtract from .
Step 6.3.2
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 6.3.3
Divide each term in by and simplify.
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Step 6.3.3.1
Divide each term in by .
Step 6.3.3.2
Simplify the left side.
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Step 6.3.3.2.1
Cancel the common factor of .
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Step 6.3.3.2.1.1
Cancel the common factor.
Step 6.3.3.2.1.2
Divide by .
Step 6.3.3.3
Simplify the right side.
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Step 6.3.3.3.1
Cancel the common factor of and .
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Step 6.3.3.3.1.1
Factor out of .
Step 6.3.3.3.1.2
Cancel the common factors.
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Step 6.3.3.3.1.2.1
Factor out of .
Step 6.3.3.3.1.2.2
Cancel the common factor.
Step 6.3.3.3.1.2.3
Rewrite the expression.
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
Combine the numerators over the common denominator.
Step 6.4.2.1.2
Multiply .
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Step 6.4.2.1.2.1
Combine and .
Step 6.4.2.1.2.2
Multiply by .
Step 6.4.2.1.3
To write as a fraction with a common denominator, multiply by .
Step 6.4.2.1.4
Combine and .
Step 6.4.2.1.5
Combine the numerators over the common denominator.
Step 6.4.2.1.6
Simplify the numerator.
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Step 6.4.2.1.6.1
Multiply by .
Step 6.4.2.1.6.2
Add and .
Step 6.4.2.1.7
Multiply the numerator by the reciprocal of the denominator.
Step 6.4.2.1.8
Cancel the common factor of .
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Step 6.4.2.1.8.1
Factor out of .
Step 6.4.2.1.8.2
Cancel the common factor.
Step 6.4.2.1.8.3
Rewrite the expression.
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