Ax + by + c

Definition

The expression 'ax + by + c' represents the general form of a linear equation in two variables, x and y. This format is crucial for determining the relationships between lines, particularly when analyzing parallel and perpendicular lines. The coefficients 'a' and 'b' relate to the slope and orientation of the line, while 'c' indicates the line's vertical position on the graph.

5 Must Know Facts For Your Next Test

1. 'ax + by + c = 0' is an equivalent representation of the general linear equation that helps identify line properties.
2. To find the slope from 'ax + by + c', rearrange it into slope-intercept form, leading to the formula 'y = -\frac{a}{b}x - \frac{c}{b}'.
3. Two lines are parallel if their coefficients 'a' and 'b' produce the same slope when transformed into slope-intercept form.
4. Two lines are perpendicular if the product of their slopes equals -1, which can be derived from 'ax + by + c' through rearranging to find respective slopes.
5. The value of 'c' in the equation affects where the line crosses the y-axis, impacting its overall position on the graph.

Review Questions

• How can you derive the slope of a line from the equation ax + by + c = 0?
  • To derive the slope from 'ax + by + c = 0', first rearrange the equation into slope-intercept form. You do this by isolating y: 'by = -ax - c', which simplifies to 'y = -\frac{a}{b}x - \frac{c}{b}'. The slope of the line is given by '-\frac{a}{b}', showing how changes in 'a' and 'b' affect the angle and steepness of the line.
• What conditions must be met for two lines represented by ax + by + c and dx + ey + f to be parallel or perpendicular?
  • For two lines to be parallel, their slopes must be equal. This means that after transforming both equations into slope-intercept form, they should yield identical values for '-\frac{a}{b}' and '-\frac{d}{e}'. Conversely, for lines to be perpendicular, their slopes must multiply to -1, leading to the condition that '-\frac{a}{b} \cdot -\frac{d}{e} = -1', indicating their intersection forms a right angle.
• Evaluate how changing the value of 'c' affects the graph of a line represented by ax + by + c.
  • 'C' represents the vertical shift of the line on a graph. Altering its value while keeping 'a' and 'b' constant will move the line up or down without changing its slope. For instance, increasing 'c' shifts the line upwards, while decreasing it shifts it downwards. This change does not affect whether the line is parallel or perpendicular to another; it merely adjusts its position relative to the y-axis.

Related terms

Slope-Intercept Form: A way of expressing linear equations in the form y = mx + b, where m represents the slope and b represents the y-intercept.

Perpendicular Lines: Lines that intersect at a right angle (90 degrees) and have slopes that are negative reciprocals of each other.

Parallel Lines: Lines that run in the same direction and never intersect, having identical slopes.