Question 925·Medium·Math

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The ordered pair $(x, y)$ satisfies the system

\begin{cases} x + y = 7 \\ xy = 10 \end{cases}

If $x > y$ , what is the value of $x$ ?

Your answer

In a nonlinear system, at least one equation doesn't graph as a line, so the graphs may meet more than once. Graph both equations in Desmos and click every intersection to find the points that satisfy both. Use the inequality to choose the point before reporting the requested coordinate. Reversing the coordinates can preserve the sum and product while failing the inequality.

Hints

Hint 1
An intersection is a point on both graphs, so its coordinates satisfy both equations. The product $xy$ makes one graph a curve. Could the line and curve meet at more than one point?
Hint 2
In an ordered pair $(x,y)$ , the first coordinate is $x$ and the second is $y$ . After you click both intersections, which point has its first coordinate greater than its second?

Step-by-step

Approach 1: Graph both equations

Step 1
Find the points that satisfy both equations
Type $x+y=7$ and $xy=10$ on separate lines in Desmos, then click both intersections. Desmos shows $(2,5)$ and $(5,2)$ . Each point lies on both graphs, so each satisfies the two equations. Keep both until you check the condition.
Step 2
Choose the point with x greater than y
At $(2,5)$ , $x=2$ is less than $y=5$ , so that point fails $x>y$ . At $(5,2)$ , $x=5$ is greater than $y=2$ . The question asks for the $x$ -coordinate, the first number, so $x=5$ . Grid in 5.

Approach 2: Find a pair with the given sum and product

Step 1
Find the two numbers
No Desmos needed. A small factor pair gives both numbers. A factor pair of $10$ is two numbers whose product is $10$ . The pair $2$ and $5$ also has the required sum: $2\times5=10$ and $2+5=7$ . Either order satisfies the two equations.
Step 2
Assign the larger number to x
Because $x>y$ , assign the larger number to $x$ : $x=5$ and $y=2$ . Putting $2$ first would fail the inequality, so the requested value of $x$ is $5$ . Grid in 5.

Question 925·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this