In Exercises 43-48, (a) find the slope and y-intercept (if...

In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).
$3 x+7=0$

Key Concepts

Graphical Representation and Verification

Graphing lines by hand involves plotting key points such as intercepts and using the slope to draw the line accurately. Using graphing utilities or software to verify the algebraic solutions aids in confirming the accuracy of the analysis and helps solidify understanding of the relationship between the algebraic equation and its graphical representation.

Vertical Lines

Vertical lines are characterized by having a constant x-value for all y-values, and therefore their equations are of the form x = constant. Such lines have an undefined slope and generally do not have a y-intercept, as they do not cross the y-axis unless the constant is zero.

Slope and y-Intercept

The slope indicates the steepness and direction of a line, typically defined as the ratio of the vertical change to the horizontal change between any two points on the line. The y-intercept is the point where the line crosses the y-axis. For many lines, the equation can be rearranged to identify these values directly, but special cases like vertical lines may not have a defined slope or y-intercept.

Linear Equation Forms

Linear equations can be expressed in various forms including the slope-intercept form (y = mx + b) and the standard form (Ax + By = C). Understanding these forms helps in identifying key characteristics such as the slope and y-intercept, or recognizing special lines such as vertical or horizontal lines.

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