Systems of Equations and Inequalities

A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where the graphs of the equations intersect.

A consistent system of equations has at least one solution, meaning the lines intersect at one or more points.

An inconsistent system of equations has no solutions, meaning the lines are parallel and never intersect.

A linear inequality is an inequality that involves a linear function. It can be written in the form $$ax + by < c$$ , $$ax + by > c$$ , etc.

The graphical representation of a linear inequality is a half-plane divided by a boundary line, which can be solid (inclusive) or dashed (exclusive).

The slope of a line represents the rate of change of the y-coordinate with respect to the x-coordinate. It indicates how steep the line is.

The slope (m) can be calculated using the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ .

The y-intercept is the point where the line crosses the y-axis, represented as (0, b) in the equation $$y = mx + b$$ .

A solid line indicates that the points on the line are included in the solution (≥ or ≤), while a dashed line indicates that the points on the line are not included (> or <).

The standard form of a linear equation is $$Ax + By = C$$ , where A, B, and C are integers, and A should be non-negative.