Question 167·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The system of equations

\begin{aligned} x+y &= 5 \\[1.4em] x^2+y^2 &= 13 \end{aligned}

has real solutions with $x \ge y$ . What is the value of $x$ ?

Your answer

A nonlinear system has two equations with the same unknowns, including at least one squared term. Its graphs can meet more than once, so type both equations into Desmos and click every intersection. Each crossing solves both equations. Then use the given inequality to choose a point, and report only the coordinate the question asks for.

Hints

Hint 1
A solution to a system is a pair $(x,y)$ that makes both equations true. Graph the first equation, then look for points on that graph that satisfy the second equation too.
Hint 2
An intersection lies on both graphs. Once you have the crossing points, compare $x$ and $y$ within each point to see which one meets $x \ge y$ .

Step-by-step

Graph the system and choose the right intersection
Step 1
Graph the first equation
Type $x+y=5$ . Desmos draws a line of points whose coordinates add to $5$ . Keep it on screen: a solution must lie on both graphs, not only this one.
Step 2
Find both crossings and apply the condition
Add $x^2+y^2=13$ and click both intersections. Desmos shows $(2,3)$ and $(3,2)$ ; each point solves the system. Compare $x$ and $y$ within the same point. At $(2,3)$ , $2<3$ , so it fails $x\ge y$ . At $(3,2)$ , $3\ge2$ . The question asks for $x$ , the first coordinate, so Grid in 3.

Question 167·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