Question

Can the equation of every line be written in slope-intercept form? Why?

Short answer

Not every line can be written in slope-intercept form. Vertical lines cannot be expressed this way due to their undefined slope.

Step-by-step solution

- Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula: \[ y = mx + b \] where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis.

- Identify the Characteristics of Lines

Recall that there are different types of lines, including horizontal and vertical lines. Horizontal lines have a slope of 0 and are in the form \( y = b \). Vertical lines, however, have an undefined (infinite) slope and are in the form \( x = a \).

- Consider Horizontal Lines

Horizontal lines can be written in slope-intercept form. For example, a horizontal line passing through \( y = 3 \) can be written as \( y = 0x + 3 \), which simplifies to \( y = 3 \).

- Analyze Vertical Lines

Vertical lines cannot be written in slope-intercept form because their slope is undefined. For example, the line \( x = 2 \) does not fit the structure of \( y = mx + b \) because it represents an infinite slope.

- Conclusion

Not every line can be written in slope-intercept form. While horizontal lines fit into the slope-intercept form, vertical lines, due to their undefined slope, cannot be expressed this way.

Key concepts

Linear Equations

A linear equation represents a straight line when plotted on a graph. The general form of a linear equation is: \[ Ax + By = C \]where \(A\), \(B\), and \(C\) are constants. Linear equations are fundamental in algebra because they describe relationships where the rate of change is constant. To analyze and graph these equations more easily, we often convert them into slope-intercept form. This conversion reveals important information about the line, such as its slope and y-intercept.

Slope of a Line

The slope of a line measures its steepness and direction. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line:\[ m = \frac{\Delta y}{\Delta x} \]When a line moves upward from left to right, its slope is positive. When it moves downward, its slope is negative. A slope of zero indicates a horizontal line, while an undefined slope corresponds to a vertical line. The slope-intercept form \( y = mx + b \) makes it easy to determine the slope (\(m\)) and the y-intercept (\(b\)), which is the point where the line crosses the y-axis.

Vertical and Horizontal Lines

Horizontal lines run left to right and have a constant y-value. Their slope is zero, and their equation is of the form: \[ y = b \] where \(b\) is the y-intercept. For example, the line \( y = 3 \) is a horizontal line crossing the y-axis at 3. These lines fit perfectly into the slope-intercept form. However, vertical lines are different. They run up and down and have a constant x-value, represented by the equation: \[ x = a \] where \(a\) is where the line crosses the x-axis. Unlike horizontal lines, vertical lines do not have a slope-intercept form because their slope is undefined. For example, the line \( x = 2 \) is a vertical line crossing the x-axis at 2. Since undefined slopes can't be represented in \( y = mx + b \), vertical lines are the exception to the slope-intercept form.