Question 158·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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For real numbers $x$ and $y$ , the system of equations

x + y = 5\qquad\text{and}\qquad x^2 + y^2 = 13

is satisfied. What is the value of $xy$ ?

A system is a set of equations that the same pair $(x,y)$ must satisfy. When one equation graphs as a line and the other as a circle, enter both in Desmos and click every intersection. Then use a point to calculate what the question asks for. A given sum is not the requested product, though swapping the two coordinates leaves their product unchanged.

Hints

Hint 1
A solution to a system makes both equations true. On a graph, that means the point lies on both graphs. Where do the line and circle meet when you enter the two equations in Desmos?
Hint 2
An ordered pair $(x,y)$ gives you two coordinates. The question asks for their product, not either coordinate or their sum. Would swapping the coordinates change that product?

Step-by-step

Approach 1: Graph the system in Desmos
Step 1
Find the points that satisfy both equations
Type $x+y=5$ and $x^2+y^2=13$ on separate lines. Desmos graphs a line and a circle. Click both intersections to read $(2,3)$ and $(3,2)$ . An intersection lies on both graphs, so either pair satisfies the system.
Step 2
Multiply the coordinates
Use $(2,3)$ : type $x_1=2$ , $y_1=3$ , and then $x_1y_1$ . Desmos displays $6$ . The other point swaps the coordinates, which doesn't change their product. Don't submit $5$ : that's the given sum $x+y$ , not $xy$ . So $xy=6$ . Choice C.

Approach 2: Link the sum directly to the product
Step 1
Square the known sum
Use the square-of-a-sum identity: $(x+y)^2=x^2+y^2+2xy$ . The $2xy$ comes from multiplying $x$ by $y$ and $y$ by $x$ . So squaring the known sum reveals the missing product without finding either coordinate.
Step 2
Replace the two known quantities
The given sum is $5$ , and the given sum of squares is $13$ . Put both values into the identity: $5^2=13+2xy$ .
Step 3
Isolate twice the product
Subtract $13$ from both sides: $5^2-13=2xy$ . Keep the $2$ attached to $xy$ : the squared sum contains two copies of the product.
Step 4
Calculate the product
Divide by $2$ : $xy=\frac{5^2-13}{2}$ . Type $\frac{5^2-13}{2}$ in Desmos; it displays $6$ . So the value of $xy$ is $6$ . Choice C.

Question 158·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