Simplify the sum of the two algebraic fractions: (3)/(x-y) + (1)/(x+y)

Question

ScanSolve AI Tutor · Accepted Answer

It looks like you're working on simplifying algebraic fractions. Based on the handwritten work below the initial expression, I will assume the problem is to add the two fractions, not multiply them.

Identify the expression to be simplified.
The problem is to simplify the sum of the two algebraic fractions:
$\frac{3}{x-y} + \frac{1}{x+y}$
Find a common denominator for the fractions.
The denominators are $(x-y)$ and $(x+y)$ . The least common denominator (LCD) is the product of these two distinct factors: $(x-y)(x+y)$ .
Rewrite each fraction with the common denominator.
To rewrite the first fraction, multiply its numerator and denominator by $(x+y)$ :
$\frac{3}{x-y} = \frac{3(x+y)}{(x-y)(x+y)}$
To rewrite the second fraction, multiply its numerator and denominator by $(x-y)$ :
$\frac{1}{x+y} = \frac{1(x-y)}{(x+y)(x-y)}$
Add the numerators over the common denominator.
$\frac{3(x+y)}{(x-y)(x+y)} + \frac{1(x-y)}{(x-y)(x+y)} = \frac{3(x+y) + 1(x-y)}{(x-y)(x+y)}$
Expand and simplify the numerator.
$3(x+y) + 1(x-y) = 3x + 3y + x - y$
Combine the like terms:
$(3x + x) + (3y - y) = 4x + 2y$
Simplify the denominator.
The denominator is $(x-y)(x+y)$ , which is a difference of squares:
$(x-y)(x+y) = x^2 - y^2$
Write the final simplified expression.
Combine the simplified numerator and denominator:
$\frac{4x+2y}{x^2-y^2}$
The numerator can also be factored as $2(2x+y)$ .
$\frac{2(2x+y)}{x^2-y^2}$

Simplify the sum of the two algebraic fractions: (3)/(x-y) + (1)/(x+y)

Answer

Step-by-step solution

Identify the expression to be simplified.

Find a common denominator for the fractions.

Rewrite each fraction with the common denominator.

Add the numerators over the common denominator.

Expand and simplify the numerator.

Simplify the denominator.

Write the final simplified expression.

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