Graphs of Linear Equations and Inequalities

INTRODUCTION

Graphs of linear equations and inequalities are an important part of algebra as they allow for the visual representation of mathematical expressions and their solutions. Graphs of linear equations and inequalities serve as a clear form of notation where the values of variables are displayed on a coordinate system. With the help of graphs of linear equations and inequalities, we can accurately depict relationships between data and identify all solutions to given expressions.

GRAPHS OF LINEAR EQUATIONS

Graphs of linear equations show all the points for which the value of the expression on the left side of the equation is equal to the value on the right side. Linear equations written in the form
y = kx + n (or y = mx + b in common English notation)
define a straight line in the coordinate system. The coefficient 'k' (or 'm') represents the direction or slope of the line, and 'n' (or 'b') represents the y-intercept, where the line crosses the vertical axis.

Graphs of linear equations are always straight lines because the variables in the equations do not have powers higher than one. Every point on the line represents a solution to the equation because, for all these pairs of x and y values, the equality from the equation holds true.

When drawing graphs of linear equations, we typically determine two arbitrary values for x, calculate the corresponding y values (or find two points like the x and y-intercepts), and plot these points in the coordinate system. Then, we draw a straight line through these points, which illustrates all solutions to the equation.

GRAPHS OF LINEAR INEQUALITIES

Graphs of linear inequalities show regions in the coordinate system where the conditions of the given inequality are met. Linear inequalities written in forms like
y > kx + n
or
y ≤ kx + n
define a part of the plane that lies either above or below the line (or to one side of a vertical line).

When graphing linear inequalities, we first draw the boundary line of the equation y = kx + n.

• If the inequality sign is strict (> or <), the line is drawn as a dashed line, as the points on this line are not included in the solution.
• If the inequality sign is inclusive (≥ or ≤), the line is drawn as a solid line, as the points on the line are part of the solution set.

After drawing the line, we determine which side of the plane corresponds to the solutions of the linear inequality. The region above the line typically means greater y-values, and the region below the line means smaller y-values (this depends on the inequality sign and whether y is isolated). This region is graphically marked by shading, which shows all solutions to the inequality. A test point can be used to determine which region to shade.

CONNECTION BETWEEN GRAPHS OF LINEAR EQUATIONS AND INEQUALITIES

Graphs of linear equations and inequalities are interconnected, as in both cases, we start by determining the boundary line, which is based on the same form of notation. The difference arises in that the graph of a linear equation represents only the points on the line, while the graph of a linear inequality includes the entire region where the values of the variables are greater than or less than (or equal to) certain boundary values.

Understanding graphs of linear equations and inequalities allows for easier representation of solutions and better insight into the relationships between expressions, as we can quickly identify correct values and ranges of variables.

CONCLUSION

Graphs of linear equations and inequalities are key in displaying the solutions of mathematical expressions on a coordinate system. Graphs of linear equations and inequalities provide a clear overview of numerical connections, determine correct value regions, and serve as a tool for precise data processing. With a proper understanding of graphs of linear equations and inequalities, we achieve greater orderliness in solving tasks where related variables appear.