Question
Which of the following equations represents a linear relationship that is also a proportional relationship? y = x2 y = 4x + 1
Explain:
Explain:
Answers
Answered by
GPT-4o mini
To determine which of the given equations represents a linear relationship that is also a proportional relationship, we need to consider the definitions of these types of relationships.
1. **Linear relationship**: An equation is linear if it can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. If \(b = 0\), the line passes through the origin (0,0).
2. **Proportional relationship**: A proportional relationship is a specific type of linear relationship where the y-intercept \(b = 0\). Such an equation can be expressed in the form \(y = kx\), where \(k\) is the constant of proportionality.
Now, let's analyze the given equations:
1. **\(y = x^2\)**:
- This is a quadratic equation, not linear. It does not represent a linear relationship.
- Since it is not linear, it cannot be proportional either.
2. **\(y = 4x + 1\)**:
- This is a linear equation, where the slope \(m = 4\) and the y-intercept \(b = 1\).
- Since \(b \neq 0\), it does not pass through the origin and is not a proportional relationship.
In conclusion, neither of the given equations represents a linear relationship that is also a proportional relationship.
- **Answer**: **Neither equation represents a linear relationship that is also a proportional relationship.**
1. **Linear relationship**: An equation is linear if it can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. If \(b = 0\), the line passes through the origin (0,0).
2. **Proportional relationship**: A proportional relationship is a specific type of linear relationship where the y-intercept \(b = 0\). Such an equation can be expressed in the form \(y = kx\), where \(k\) is the constant of proportionality.
Now, let's analyze the given equations:
1. **\(y = x^2\)**:
- This is a quadratic equation, not linear. It does not represent a linear relationship.
- Since it is not linear, it cannot be proportional either.
2. **\(y = 4x + 1\)**:
- This is a linear equation, where the slope \(m = 4\) and the y-intercept \(b = 1\).
- Since \(b \neq 0\), it does not pass through the origin and is not a proportional relationship.
In conclusion, neither of the given equations represents a linear relationship that is also a proportional relationship.
- **Answer**: **Neither equation represents a linear relationship that is also a proportional relationship.**