Determine whether there exists a constant c such that the...

Determine whether there exists a constant $c$ such that the line $x+c y=1:$
\begin{equation}
\begin{array}{ll}{\text { (a) has slope 4. }} & {\text { (b) passes through }(3,1).} \\ {\text { (c) is horizontal. }} & {\text { (d) is vertical. }}\end{array}
\end{equation}

Key Concepts

Linear Equation in Two Variables

A linear equation in two variables represents a straight line in the Cartesian plane. It is typically written in forms such as Ax + By = C or y = mx + b, where A, B, and C (or m and b) are constants. Understanding this form helps in determining the various properties of the line, such as its slope, intercepts, and orientation in the plane.

Slope and Slope-Intercept Form

The slope of a line is a measure of its steepness and is usually denoted by m in the slope-intercept form y = mx + b. Converting a linear equation into this form reveals the slope directly, which is essential for analyzing the line's direction and behavior. It also aids in comparing lines based on how rapidly they rise or fall.

Horizontal Lines

A horizontal line is a special case of a linear equation that has a slope of zero. This means that no matter the change in x, the value of y remains constant. Recognizing when a linear equation represents a horizontal line is important, as it has specific characteristics, such as a constant y-value and a graph that is parallel to the x-axis.

Vertical Lines

A vertical line is another special case where the line is parallel to the y-axis. Here, the slope is undefined, and all points on the line have the same x-coordinate. Understanding vertical lines is crucial because their standard form cannot be easily rewritten in the slope-intercept form, requiring a different method of analysis.

Point-Substitution in Linear Equations

Verifying whether a particular point lies on a given line involves substituting the coordinates of the point into the line's equation. If the equation is satisfied, the point is on the line. This technique is fundamental in coordinate geometry for checking solutions and for determining parameters that align the line with specific points.

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