If x + y = 10 and x - y = 4, what is the value of xy? | Mathematics MCQ | Examoogle

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If x + y = 10 and x - y = 4, what is the value of xy?

Correct Answer

📐 MA • Math Direct FormulaMathematicsAlgebra

Option A

Quick Summary:

Use the identity $xy = \frac{(x+y)^2 - (x-y)^2}{4}$ . Substituting the values, we get $(10^2 - 4^2) / 4 = (100 - 16) / 4 = 84 / 4 = 21$ .

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📐MAMath SolutionDirect Formula

📋 Given

We are given the sum of two variables x + y = 10 and their difference x - y = 4.

🔢 Formula Used

xy=(x+y2)2−(x−y2)2xy = \left(\frac{x+y}{2}\right)^2 - \left(\frac{x-y}{2}\right)^2xy=(2x+y​)2−(2x−y​)2

⚡ Exam Hall Shortcut / Speed Trick

Use the identity $xy = \frac{(x+y)^2 - (x-y)^2}{4}$ . Substituting the values, we get $(10^2 - 4^2) / 4 = (100 - 16) / 4 = 84 / 4 = 21$ .

⚠️ Common Student Trap / Pitfall

Many students try to solve for $x$ and $y$ individually (which is $x=7, y=3$ ) but often make arithmetic errors in the multiplication step instead of using the algebraic identity.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify the relevant identity

To find the product $xy$ when sum and difference are known, we use the identity based on the square of sum and difference.

xy=(x+y)2−(x−y)24xy = \frac{(x+y)^2 - (x-y)^2}{4}xy=4(x+y)2−(x−y)2​

Substitute the given values

Substitute $x+y = 10$ and $x-y = 4$ into the formula.

xy=(10)2−(4)24xy = \frac{(10)^2 - (4)^2}{4}xy=4(10)2−(4)2​

Calculate the squares

Calculate $10^2 = 100$ and $4^2 = 16$ .

xy=100−164xy = \frac{100 - 16}{4}xy=4100−16​

Final simplification

Subtract the values and divide by 4 to get the product $xy = 84/4 = 21$ .

xy=21xy = 21xy=21

✅

A is correct because the calculation $(100 - 16)/4$ yields 21.

Core Concepts Used

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Algebraic Identities System of Linear Equations Difference of Squares

💡 EXAM TIP

This identity is a specific case of the polar form of complex numbers or coordinate geometry distance formulas; mastering it saves time in geometry problems involving squares and rectangles.

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