Let x y = x + y + 10. Find (a) the identity element and (b) the inverses of 3 and −5.

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ScanSolve AI Tutor · Accepted Answer

Here's the solution for question 1: 1. (a) Identity element Step 1: Define the identity element e. For a binary operation x Δ y = x + y + 10, the identity element e is such that x Δ e = x for all x ∈ R. Step 2: Solve for e. x + e + 10 = x; e + 10 = 0; e = −10 The identity element is −10. 1. (b) Inverses of 3 and −5 Step 1: Define the inverse element. The inverse of an element a, denoted a⁻¹, is such that a Δ a⁻¹ = e. We found e = −10. Step 2: Find the inverse of 3. Let the inverse of 3 be a. 3 Δ a = e; 3 + a + 10 = −10; a + 13 = −10; a = −10 − 13; a = −23 The inverse of 3 is −23. Step 3: Find the inverse of −5. Let the inverse of −5 be b. −5 Δ b = e; −5 + b + 10 = −10; b + 5 = −10; b = −10 − 5; b = −15 The inverse of −5 is −15.

Let x y = x + y + 10. Find (a) the identity element and (b) the inverses of 3 and -5.

Step-by-step solution

Define the identity element $e$ .

Solve for $e$ .

Define the inverse element.

Find the inverse of 3.

Find the inverse of -5.

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