If ax = b, by = c and cz = a, prove that: xyz = 1



This solution deals with factoring binomials using the difference of squares.

Step by Step Solution



Rearrange the equation by subtracting what is lớn the right of the equal sign from both sides of the equation : x*y*z-(1/x*y*z)=0

Step by step solution :

Step 1 :

1 Simplify — xEquation at the over of step 1 : 1 xyz - ((— • y) • z) = 0 x

Step 2 :

Rewriting the whole as an Equivalent Fraction :2.1Subtracting a fraction from a whole Rewrite the whole as a fraction using x as the denominator :

xyz xyz • x xyz = ——— = ——————— 1 x Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction và the other fraction involved in the calculation cốt truyện the same denominator

Adding fractions that have a common denominator :2.2 Adding up the two equivalent fractions showroom the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce khổng lồ lowest terms if possible:

xyz • x - (yz) x2yz - yz —————————————— = ————————— x x

Step 3 :

Pulling out lượt thích terms :3.1 Pull out like factors:x2yz - yz=yz•(x2 - 1)

Trying lớn factor as a Difference of Squares:3.2 Factoring: x2 - 1 Theory : A difference of two perfect squares, A2-B2can be factored into (A+B)•(A-B)Proof:(A+B)•(A-B)= A2 - AB+BA-B2= A2 -AB+ AB - B2 = A2 - B2Note : AB = cha is the commutative property of multiplication. Lưu ý : -AB+ AB equals zero and is therefore eliminated from the expression.Check: 1 is the square of 1Check: x2 is the square of x1Factorization is :(x + 1)•(x - 1)

Equation at the end of step 3 :

yz • (x + 1) • (x - 1) —————————————————————— = 0 x

Step 4 :

When a fraction equals zero :4.1 When a fraction equals zero ...Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.Here"s how:

yz•(x+1)•(x-1) —————————————— • x = 0 • x x Now, on the left hand side, the x cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:yz • (x+1) • (x-1)=0

Theory - Roots of a hàng hóa :4.2 A product of several terms equals zero.When a sản phẩm of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going to solve as many equations as there are terms in the productAny solution of term = 0 solves sản phẩm = 0 as well.

Solving a Single Variable Equation:4.3Solveyz=0 Setting any of the variables khổng lồ zero solves the equation:y=0z=0

Solving a Single Variable Equation:

4.4Solve:x+1 = 0Subtract 1 from both sides of the equation:x = -1

Solving a Single Variable Equation:4.5Solve:x-1 = 0Add 1 to both sides of the equation:x = 1