Solve for x in terms of a and b: a+b+x1=a1+b1+x1(Where a, b, x ≠ 0 and a + b + x ≠ 0)
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Step-by-Step Solution
Step 1: Rearrange the equation
To begin solving for x, we first rearrange the equation by moving the term x1 from the right side to the left side. This groups terms involving x together.
Step 2: Combine fractions on both sides
Next, we combine the fractions on both sides of the equation by finding a common denominator for each side. On the left, the common denominator is x(a+b+x), and on the right, it is ab.
Step 3: Simplify the equation
We simplify the numerator on the left side: x−(a+b+x)=x−a−b−x=−a−b. This gives us x(a+b+x)−(a+b)=aba+b.
Step 4: Cross-multiply and solve for x
Since a+b=0 (otherwise the original equation would be undefined), we can divide both sides by (a+b). This simplifies the equation to x(a+b+x)−1=ab1. Then, we cross-multiply to get −ab=x(a+b+x).
Step 5: Form a quadratic equation
We expand the right side of the equation: x(a+b+x)=x2+(a+b)x. Moving −ab to the right side, we get x2+(a+b)x+ab=0. This is a quadratic equation in terms of x.
Step 6: Factor the quadratic equation
We can factor the quadratic equation x2+(a+b)x+ab=0 into (x+a)(x+b)=0. This is because a×b=ab and a+b=a+b.
Step 7: Solve for x
From the factored form (x+a)(x+b)=0, we set each factor equal to zero to find the possible values for x. This gives us x+a=0⟹x=−a or x+b=0⟹x=−b.