Calculate the Product: Solve (x+y)² = 1 and (x²+y²)/(x+y)²=3

Q: Why can't I just solve for x and y individually?

The system has infinitely many solutions for x and y! For example, x = 1.5 and y = -0.5, or x = 0 and y = ±1 both satisfy the conditions. The problem specifically asks for the product xy, which has a unique value.

Q: How do I remember the algebraic identity?

Think of (x+y)² as a square: you get x², y², and two rectangles of area xy each. So (x+y)² = x² + 2xy + y². Practice expanding a few examples like (3+4)² = 9 + 24 + 16.

Q: What if I get confused with the substitution steps?

Write each step clearly! From $ \frac{x^2+y^2}{(x+y)^2}=3 $, multiply both sides by (x+y)² to get x² + y² = 3(x+y)². Then substitute the known value (x+y)² = 1.

Q: Are there other ways to solve this problem?

You could use the identity x² + y² = (x+y)² - 2xy directly! Substituting: 3 = 1 - 2xy, so 2xy = 1 - 3 = -2, giving xy = -1. Same answer, fewer steps!

Algebraic Identity with Substitution Method

$(x+y)^2=1,\frac{x^2+y^2}{(x+y)^2}=3$

Calculate the product of x and y.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the value of the product XY

00:03 Substitute the first equation into the second

00:21 Add the term 2XY to each side

00:36 Use the shortened multiplication formulas to find the brackets

00:43 Substitute the first equation again

00:49 Isolate XY

01:06 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(x+y)^2=1,\frac{x^2+y^2}{(x+y)^2}=3$

Calculate the product of x and y.

Step-by-step solution

To solve the given problem, we will proceed with the following steps:

Step 1: Apply the identity for the square of the sum.
Step 2: Use the given equation to solve for the variables.
Step 3: Derive the product $xy$ .

Step 1: Using the identity
For $(x+y)^2 = x^2 + 2xy + y^2$ , we know from the problem that $(x+y)^2 = 1$ , so:

$x^2 + 2xy + y^2 = 1$

Step 2: Utilizing the second given equation,
We have $\frac{x^2+y^2}{(x+y)^2}=3$ . Therefore:

$x^2 + y^2 = 3(x+y)^2 = 3 \times 1 = 3$

Step 3: Substitute $x^2+y^2 = 3$ into the identity:

$3 + 2xy = 1$

Solving for $xy$ , we get:

$2xy = 1 - 3 = -2$

$xy = -1$

Therefore, the product of $x$ and $y$ is $\mathbf{-1}$ .

Final Answer

$xy=-1$

Key Points to Remember

Essential concepts to master this topic

Identity: Use (x+y)² = x² + 2xy + y² to expand expressions
Technique: From x² + y² = 3(x+y)² = 3(1) = 3
Check: Substitute xy = -1: 3 + 2(-1) = 1 ✓

Common Mistakes

Avoid these frequent errors

Trying to solve for individual x and y values
Don't attempt to find x and y separately = creates unnecessary complexity! The problem only asks for xy, not individual values. Always use the algebraic identity to relate (x+y)² and x² + y² directly.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why can't I just solve for x and y individually?

The system has infinitely many solutions for x and y! For example, x = 1.5 and y = -0.5, or x = 0 and y = ±1 both satisfy the conditions. The problem specifically asks for the product xy, which has a unique value.

How do I remember the algebraic identity?

Think of (x+y)² as a square: you get x², y², and two rectangles of area xy each. So (x+y)² = x² + 2xy + y². Practice expanding a few examples like (3+4)² = 9 + 24 + 16.

What if I get confused with the substitution steps?

Write each step clearly! From $\frac{x^2+y^2}{(x+y)^2}=3$ , multiply both sides by (x+y)² to get x² + y² = 3(x+y)². Then substitute the known value (x+y)² = 1.

How can I verify my answer is correct?

Substitute xy = -1 back into the identity: x² + 2xy + y² = 1 becomes x² + 2(-1) + y² = x² + y² - 2. Since x² + y² = 3, we get 3 - 2 = 1 ✓

Are there other ways to solve this problem?

You could use the identity x² + y² = (x+y)² - 2xy directly! Substituting: 3 = 1 - 2xy, so 2xy = 1 - 3 = -2, giving xy = -1. Same answer, fewer steps!

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