Question
What is the proportional relationship. Derive the equation of the line y=mx through the origin.
Answers
GPT-4o mini
A proportional relationship between two quantities means that one quantity is a constant multiple of the other. In mathematical terms, if \( y \) is proportional to \( x \), we can express this relationship as:
\[
y = kx
\]
where \( k \) is the constant of proportionality.
When the relationship is specifically through the origin (meaning that when \( x = 0 \), \( y \) also equals \( 0 \)), the equation takes the form \( y = mx \), where \( m \) is the slope of the line. This equation indicates that for every unit increase in \( x \), \( y \) increases by \( m \) units.
### Deriving the equation \( y = mx \)
1. **Understanding the slope**: The slope \( m \) represents the rate of change of \( y \) with respect to \( x \). It defines how steep the line is.
\[
m = \frac{\Delta y}{\Delta x}
\]
where \( \Delta y \) is the change in \( y \) and \( \Delta x \) is the change in \( x \).
2. **Starting from proportional relationship**: If we know that \( y \) is proportional to \( x \), we can write:
\[
y = kx
\]
Since we want the line to go through the origin, it simplifies to the case where the constant \( k \) can be replaced by \( m \), which gives us the linear equation:
\[
y = mx
\]
3. **Conclusion**: This equation is linear and passes through the origin (0,0). It illustrates the concept of proportionality where \( y \) varies directly with \( x \), multiplied by the constant \( m \).
### Example:
If \( m = 2 \), the equation \( y = 2x \) means that for every increase of 1 in \( x \), \( y \) increases by 2. If \( x = 3 \), then:
\[
y = 2 \cdot 3 = 6
\]
Thus, the relationship is consistent with the proportional relationship. In summary, the equation \( y = mx \) describes a proportional relationship through the origin.
\[
y = kx
\]
where \( k \) is the constant of proportionality.
When the relationship is specifically through the origin (meaning that when \( x = 0 \), \( y \) also equals \( 0 \)), the equation takes the form \( y = mx \), where \( m \) is the slope of the line. This equation indicates that for every unit increase in \( x \), \( y \) increases by \( m \) units.
### Deriving the equation \( y = mx \)
1. **Understanding the slope**: The slope \( m \) represents the rate of change of \( y \) with respect to \( x \). It defines how steep the line is.
\[
m = \frac{\Delta y}{\Delta x}
\]
where \( \Delta y \) is the change in \( y \) and \( \Delta x \) is the change in \( x \).
2. **Starting from proportional relationship**: If we know that \( y \) is proportional to \( x \), we can write:
\[
y = kx
\]
Since we want the line to go through the origin, it simplifies to the case where the constant \( k \) can be replaced by \( m \), which gives us the linear equation:
\[
y = mx
\]
3. **Conclusion**: This equation is linear and passes through the origin (0,0). It illustrates the concept of proportionality where \( y \) varies directly with \( x \), multiplied by the constant \( m \).
### Example:
If \( m = 2 \), the equation \( y = 2x \) means that for every increase of 1 in \( x \), \( y \) increases by 2. If \( x = 3 \), then:
\[
y = 2 \cdot 3 = 6
\]
Thus, the relationship is consistent with the proportional relationship. In summary, the equation \( y = mx \) describes a proportional relationship through the origin.