# Solve by Substitution ax+by=1 , bx+ay=1

,
Solve for .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
Substitute for into then solve for .
Replace with in the equation.
Solve the equation for .
Simplify each term.
Apply the distributive property.
Combine and .
Multiply .
Combine and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factor for is itself.
occurs time.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Solve the equation.
Subtract from both sides of the equation.
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify the numerator.
Raise to the power of .
Apply the distributive property.
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Apply the distributive property.
Multiply by .
Factor using the perfect square rule.
Rewrite as .
Rewrite as .
Check the middle term by multiplying and compare this result with the middle term in the original expression.
Simplify.
Factor using the perfect square trinomial rule , where and .
Pull terms out from under the radical, assuming positive real numbers.
Simplify the expression to solve for the portion of the .
Change the to .
Simplify the numerator.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Apply the distributive property.
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Apply the distributive property.
Multiply by .
Factor using the perfect square rule.
Rewrite as .
Rewrite as .
Check the middle term by multiplying and compare this result with the middle term in the original expression.
Simplify.
Factor using the perfect square trinomial rule , where and .
Pull terms out from under the radical, assuming positive real numbers.
Change the to .
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Cancel the common factor.
Divide by .
The final answer is the combination of both solutions.
Substitute for into then solve for .
Replace with in the equation.
Solve the equation for .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.
Add to both sides of the equation.
Subtract from .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Substitute for into then solve for .
Replace with in the equation.
Solve the equation for .
Multiply by .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
The solution to the system is the complete set of ordered pairs that are valid solutions.
Solve by Substitution ax+by=1 , bx+ay=1