Linear Equations

Definition

A linear equation is a mathematical equation in which the variables are raised only to the first power and the equation forms a straight line when graphed. These equations are fundamental in algebra and have numerous applications in various fields.

5 Must Know Facts For Your Next Test

1. Linear equations can have one variable (e.g., $2x + 5 = 15$) or multiple variables (e.g., $3x + 2y = 10$).
2. The solution to a linear equation is the value of the variable that makes the equation true.
3. Linear equations can be used to model real-world situations, such as the relationship between distance, rate, and time.
4. The slope of a linear equation represents the rate of change between the dependent and independent variables.
5. Graphing linear equations can help visualize the relationship between the variables and identify the solution.

Review Questions

• Explain how linear equations can be used to model real-world situations.
  • Linear equations are often used to model linear relationships in the real world, where one variable changes at a constant rate with respect to another variable. For example, the relationship between distance, rate, and time can be represented by the linear equation $d = rt$, where $d$ is the distance traveled, $r$ is the rate of travel, and $t$ is the time. By setting up a linear equation to represent the problem, you can solve for the unknown variable, such as the time required to travel a certain distance at a given rate.
• Describe the process of graphing a linear equation and how it can help identify the solution.
  • Graphing a linear equation involves plotting the points that satisfy the equation on a coordinate plane. The resulting graph will be a straight line. The solution to the linear equation, which is the value of the variable that makes the equation true, can be identified by finding the point where the line intersects the $x$-axis or $y$-axis. This point represents the value of the variable that satisfies the equation. Graphing linear equations can also help visualize the relationship between the variables and understand the properties of the equation, such as the slope and $y$-intercept.
• Analyze how the slope of a linear equation relates to the rate of change between the variables.
  • The slope of a linear equation represents the rate of change between the dependent and independent variables. Specifically, the slope $m$ in the slope-intercept form $y = mx + b$ indicates the change in the $y$-value for a unit change in the $x$-value. This rate of change is constant throughout the entire linear equation. Understanding the slope of a linear equation is crucial in modeling real-world situations, as it allows you to quantify the relationship between the variables and make predictions about how changes in one variable will affect the other. The slope can also provide insights into the direction and steepness of the line, which can be useful in interpreting the meaning of the equation in the context of the problem.

Related terms

Slope-Intercept Form: The standard form of a linear equation written as $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Graphing Linear Equations: The process of plotting the points that satisfy a linear equation on a coordinate plane, resulting in a straight line.

Solving Linear Equations: The method of finding the value of the variable that makes the equation true, often involving algebraic manipulations.