An equation is considered proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant. In a proportional relationship, the ratio of \( y \) to \( x \) remains constant. This means that if you double \( x \), \( y \) will also double, and if you halve \( x \), \( y \) will also be halved. Additionally, the graph of a proportional relationship is a straight line that passes through the origin (0,0).
Example of a Proportional Equation:
Consider the equation \( y = 3x \).
- In this equation, \( k = 3 \).
- For every increase in \( x \), \( y \) increases by three times that amount.
- For instance:
- If \( x = 1 \), then \( y = 3(1) = 3 \)
- If \( x = 2 \), then \( y = 3(2) = 6 \)
- If \( x = 3 \), then \( y = 3(3) = 9 \)
- For instance:
- The ratio \( \frac{y}{x} = 3 \) remains constant, and the graph of this equation is a straight line that passes through the origin (0,0).
Example of a Non-Proportional Graph:
Consider a non-proportional equation like \( y = 2x + 1 \).
- This equation has a y-intercept of 1, which means it does not pass through the origin.
- As \( x \) increases, the \( y \) value increases, but because of the constant term (+1), the ratio of \( y \) to \( x \) is not constant:
- For example:
- If \( x = 1 \), then \( y = 2(1) + 1 = 3 \) → \( \frac{y}{x} = 3 \)
- If \( x = 2 \), then \( y = 2(2) + 1 = 5 \) → \( \frac{y}{x} = 2.5 \)
- If \( x = 3 \), then \( y = 2(3) + 1 = 7 \) → \( \frac{y}{x} = \frac{7}{3} \approx 2.33 \)
- For example:
- The ratios are not consistent, meaning the graph of this equation is a straight line that does not pass through the origin, indicating a non-proportional relationship.
In summary, a proportional equation is characterized by direct variation where the graph is linear and passes through (0,0), while a non-proportional equation usually has a constant added that shifts the line vertically, not passing through the origin.