Linear Equations

Definition

A linear equation is a mathematical equation in which the variables are raised to the first power and the equation can be represented as a straight line on a graph. These equations are fundamental in solving systems of equations and graphing systems of linear inequalities.

5 Must Know Facts For Your Next Test

1. Linear equations can be used to model and solve real-world problems, such as finding the cost of a certain number of items or the time it takes to travel a certain distance.
2. The solution to a system of linear equations is the point where the lines intersect, and this solution satisfies all the equations in the system.
3. Graphing a system of linear inequalities involves plotting the boundary lines and shading the region where all the inequalities are true.
4. The slope of a linear equation represents the rate of change between the variables, and the $y$-intercept is the point where the line crosses the $y$-axis.
5. Linear equations can be classified as consistent (have a unique solution), inconsistent (have no solution), or dependent (have infinitely many solutions).

Review Questions

• Explain how linear equations can be used to solve applications with systems of equations.
  • Linear equations are the foundation for solving systems of equations, which are sets of two or more linear equations with the same variables. By finding the point where the lines representing the equations intersect, you can determine the values of the variables that satisfy all the equations in the system. This allows you to solve real-world problems, such as finding the cost of a certain number of items or the time it takes to travel a certain distance, by setting up a system of linear equations and solving for the unknown values.
• Describe the process of graphing a system of linear inequalities.
  • Graphing a system of linear inequalities involves plotting the boundary lines, which are linear equations, and shading the appropriate half-plane based on the inequality symbol. The region where all the inequalities are true is the solution set for the system. This process allows you to visualize the feasible region and make decisions based on the constraints represented by the linear inequalities. For example, you might use a system of linear inequalities to represent the production constraints and profit maximization for a manufacturing company.
• Evaluate how the characteristics of linear equations, such as slope and $y$-intercept, impact the solutions to systems of equations and the graphing of linear inequalities.
  • The slope and $y$-intercept of a linear equation are crucial in determining the properties of the line and how it interacts with other lines in a system of equations or inequalities. The slope represents the rate of change between the variables, which affects the orientation and steepness of the line. The $y$-intercept is the point where the line crosses the $y$-axis, which determines the starting point of the line. These characteristics directly impact the solutions to systems of equations, as the point of intersection is determined by the slopes and $y$-intercepts of the individual equations. Similarly, the slope and $y$-intercept influence the graphing of linear inequalities, as they define the boundary lines and the direction of the shaded region. Understanding these properties of linear equations is essential for solving applications with systems of equations and graphing systems of linear inequalities.

Related terms

System of Equations:

A system of equations is a set of two or more linear equations with the same variables, where the solution satisfies all the equations simultaneously.

Slope-Intercept Form:

The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept of the line.

Graphing Linear Inequalities: Graphing linear inequalities involves plotting the boundary line, which is a linear equation, and shading the appropriate half-plane based on the inequality symbol.