By the end of the section on the slope of a line, learners will be able to find the slope of a line, graph a line given a point and its slope, graph a line using its slope and intercept, choose the most convenient method to graph a line, graph and interpret applications involving the slope-intercept form, and use slopes to identify parallel and perpendicular lines.

Claude

An equation that can be written in the form $$Ax + By = C$$, where the coefficients $$A$$ and $$B$$ are not both zero, is defined as a linear equation in two variables. This establishes a mathematical relationship between two unknown quantities, which are typically represented by variables such as $$x$$ and $$y$$.

Linear Equation in Two Variables

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OpenStax Intermediate Algebra

A linear equation involving three variables is an algebraic expression that can be written in the standard form $$ax + by + cz = d$$. In this structure, the coefficients $$a$$, $$b$$, $$c$$, and the constant $$d$$ must be real numbers, with the additional restriction that $$a$$, $$b$$, and $$c$$ cannot all simultaneously be equal to $$0$$.

Linear Equation in Three Variables

A linear equation in two variables is said to be in standard form when it is written as $$Ax + By = C$$. When writing an equation in this form, it is generally preferred—though not strictly required—that $$A$$, $$B$$, and $$C$$ are integers and that $$A \ge 0$$. Equations that do not initially appear in this format, such as $$y = -3x + 5$$, can be manipulated using properties of equality to rewrite them into the standard $$Ax + By = C$$ structure (e.g., $$3x + y = 5$$) to easily confirm they are linear equations.

Standard Form of a Linear Equation

An ordered pair $$(x, y)$$ is a solution to a linear equation in two variables, such as $$Ax + By = C$$, if substituting the $$x$$- and $$y$$-values into the equation results in a mathematically true statement. Because every substituted $$x$$-value yields a corresponding $$y$$-value that satisfies the relationship, a linear equation in two variables has infinitely many solutions.

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