Question
A system of equations can have zero, exactly one, several, or infinite solutions.
Use the description below each axis to sketch a nonlinear system of two equations with the given number of solutions.
V
x
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X
This system has choose your answer... because the two graphs never intersect.
This system has choose your answer... because the two graphs intersect at exactly one point.
Previous
This graph has choose your answer...
because
the two graphs intersect at two points.
Use the description below each axis to sketch a nonlinear system of two equations with the given number of solutions.
V
x
٧
X
This system has choose your answer... because the two graphs never intersect.
This system has choose your answer... because the two graphs intersect at exactly one point.
Previous
This graph has choose your answer...
because
the two graphs intersect at two points.
Answers
GPT-4o mini
To illustrate the different scenarios of a nonlinear system of two equations visually, let's analyze the three cases you've provided and describe how you would sketch them.
### 1. A system with **zero** solutions
- **Description**: The two graphs never intersect.
- **Example**: You can sketch a circle and a line that are positioned quite far apart so that they do not touch. For example, a circle centered at (0, 0) with a radius of 1 (equation: \(x^2 + y^2 = 1\)) and a horizontal line, such as \(y = 2\).
### 2. A system with **exactly one** solution
- **Description**: The two graphs intersect at exactly one point.
- **Example**: A circle (e.g., \(x^2 + y^2 = 1\)) intersecting a tangent line. For instance, if the line \(y = \sqrt{1 - x^2}\) is drawn, it touches the circle at the topmost point (0, 1) only.
### 3. A system with **two** solutions
- **Description**: The two graphs intersect at two points.
- **Example**: You could sketch a parabola (e.g., \(y = x^2\)) and a line (e.g., \(y = -x + 1\)). The parabola opens upwards, and the line can intersect it at two distinct points.
### Sketching Instructions
1. **For zero solutions**: Draw a circle centered at (0, 0) with radius 1 and a horizontal line above it (e.g., \(y = 2\)).
2. **For exactly one solution**: Draw the same circle and then draw a tangent line (e.g., a horizontal line just touching the top of the circle at (0, 1)).
3. **For two solutions**: Draw a parabola (which opens upwards) and a line that crosses it twice (e.g., a sloped line).
By illustrating these three scenarios in a coordinate grid, you show different behaviors of nonlinear equations and their intersections.
### 1. A system with **zero** solutions
- **Description**: The two graphs never intersect.
- **Example**: You can sketch a circle and a line that are positioned quite far apart so that they do not touch. For example, a circle centered at (0, 0) with a radius of 1 (equation: \(x^2 + y^2 = 1\)) and a horizontal line, such as \(y = 2\).
### 2. A system with **exactly one** solution
- **Description**: The two graphs intersect at exactly one point.
- **Example**: A circle (e.g., \(x^2 + y^2 = 1\)) intersecting a tangent line. For instance, if the line \(y = \sqrt{1 - x^2}\) is drawn, it touches the circle at the topmost point (0, 1) only.
### 3. A system with **two** solutions
- **Description**: The two graphs intersect at two points.
- **Example**: You could sketch a parabola (e.g., \(y = x^2\)) and a line (e.g., \(y = -x + 1\)). The parabola opens upwards, and the line can intersect it at two distinct points.
### Sketching Instructions
1. **For zero solutions**: Draw a circle centered at (0, 0) with radius 1 and a horizontal line above it (e.g., \(y = 2\)).
2. **For exactly one solution**: Draw the same circle and then draw a tangent line (e.g., a horizontal line just touching the top of the circle at (0, 1)).
3. **For two solutions**: Draw a parabola (which opens upwards) and a line that crosses it twice (e.g., a sloped line).
By illustrating these three scenarios in a coordinate grid, you show different behaviors of nonlinear equations and their intersections.