Question 219·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{aligned} x^2 + y^2 &= 10 \\[1.4em] x + y &= 4 \end{aligned}

Given the system of equations above, what is the value of $xy$ ?

A square-of-a-sum identity lets you find a product without finding the two numbers separately. The cue is a given sum and a given sum of squares. Square the sum, replace the sum of squares, and use a Desmos regression to find the product. Don’t drop the middle term $2xy$ .

Hints

Hint 1
The square of a sum has a middle term: $(x+y)^2=x^2+y^2+2xy$ . That term contains the product the question asks for, so you don’t need to find $x$ and $y$ separately.
Hint 2
Replace $x+y$ and $x^2+y^2$ with their given values. Then only the product is unknown. In Desmos, let $x_1$ stand for that product, not for the original coordinate $x$ .

Step-by-step

Square the given sum
Step 1
Expose the product
Use the square-of-a-sum identity:
$(x+y)^2=x^2+y^2+2xy.$
Multiplying $(x+y)(x+y)$ produces $xy$ twice, once from each cross-pair. Squaring a sum adds twice the product.
Step 2
Square the given sum
Since $x+y=4$ , square the whole sum in the identity:
$4^2=x^2+y^2+2xy.$
Step 3
Replace the sum of squares
The other equation says $x^2+y^2=10$ , so replace that entire sum:
$4^2=10+2xy.$
Step 4
Find the requested product
Let $x_1$ stand for $xy$ and type $4^2\sim10+2x_1$ in Desmos. A regression uses $\sim$ to find the unknown value. Under PARAMETERS, Desmos shows $x_1=3$ , so the requested product $xy$ is $3$ . Choice B.

Question 219·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this