Question 59·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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x^2 + y^2 = 25

y = x + 1

When the equations above are graphed in the $xy$ -plane, what are the coordinates $(x, y)$ of the points of intersection of the two graphs?

A system of equations asks for points that satisfy both equations. A squared $x$ plus a squared $y$ signals a circle, and a line can cross a circle more than once. Graph both equations as written in Desmos and click every intersection. Then read each point in $(x,y)$ order; a point on the circle alone may not satisfy the line.

Hints

Hint 1
A solution to a system is a point that makes both equations true. What does $y=x+1$ tell you about the second coordinate compared with the first?
Hint 2
The equation $x^2+y^2=25$ draws a circle, not a filled region. Add the line to its graph and look for every place they cross.

Step-by-step

Graph both equations and read the crossings
Step 1
Use the line to check coordinate order
Since $y=x+1$ , the second coordinate of each answer point must be $1$ greater than the first. An intersection is a point shared by two graphs, so each answer point must lie on both the circle and the line.
Step 2
Graph the circle
Type $x^2+y^2=25$ as written. Desmos draws a circle centered at $(0,0)$ that reaches $5$ units in each direction, since $25=5^2$ . Don't add $y=$ in front; that would change the equation.
Step 3
Click both intersections
Add $y=x+1$ on a new line. Click both places where the line crosses the circle. Desmos shows $(3,4)$ and $(-4,-3)$ . At each point, $y$ is $1$ greater than $x$ , and the point is on the circle. These are the requested coordinates. Choice A.

Question 59·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