Linear Equations

Definition

A linear equation is a mathematical equation in which the variables are raised only to the first power and the variables are connected by addition, subtraction, or scalar multiplication. These equations form straight lines when graphed on a coordinate plane.

5 Must Know Facts For Your Next Test

1. Linear equations can be used to model and solve a wide variety of real-world problems, such as finding the cost of an item, the speed of an object, or the amount of time it takes to complete a task.
2. The slope of a linear equation represents the rate of change between the dependent and independent variables, indicating how quickly the dependent variable changes as the independent variable changes.
3. The $y$-intercept of a linear equation is the point where the line crosses the $y$-axis, representing the value of the dependent variable when the independent variable is zero.
4. Systems of linear equations can be solved using various methods, such as substitution, elimination, or graphing, to find the unique solution that satisfies all the equations simultaneously.
5. Linear equations are fundamental in physics, as they are used to describe the relationships between physical quantities, such as displacement, velocity, and acceleration, in the context of 1.7 Solving Problems in Physics.

Review Questions

• Explain how the slope and $y$-intercept of a linear equation can be used to describe the relationship between two variables.
  • The slope of a linear equation represents the rate of change between the dependent and independent variables, indicating how quickly the dependent variable changes as the independent variable changes. The $y$-intercept represents the value of the dependent variable when the independent variable is zero. Together, the slope and $y$-intercept provide a complete description of the linear relationship between the two variables, allowing you to make predictions and solve problems involving the relationship.
• Describe the process of solving a system of linear equations and explain how it can be used to model and solve real-world problems in the context of 1.7 Solving Problems in Physics.
  • A system of linear equations consists of two or more linear equations with the same variables. To solve a system of linear equations, you can use methods such as substitution, elimination, or graphing to find the unique solution that satisfies all the equations simultaneously. In the context of 1.7 Solving Problems in Physics, systems of linear equations can be used to model and solve a variety of real-world problems, such as finding the position, velocity, or acceleration of an object, or determining the forces acting on a system. By solving the system of equations, you can determine the values of the unknown variables that describe the physical situation.
• Analyze how the properties of linear equations, such as their linearity and the ability to represent them in different forms (slope-intercept, point-slope), can be leveraged to solve complex problems in 1.7 Solving Problems in Physics.
  • The properties of linear equations, such as their linearity and the ability to represent them in different forms, are crucial in the context of 1.7 Solving Problems in Physics. The linearity of these equations allows for the use of powerful mathematical techniques, such as systems of linear equations and matrix methods, to solve complex problems involving multiple variables and constraints. The different forms of linear equations, like slope-intercept and point-slope, provide flexibility in how the equations can be expressed and manipulated, enabling you to choose the most appropriate representation for the given problem. By leveraging these properties, you can effectively model and solve a wide range of physics problems, from kinematics to dynamics, and gain a deeper understanding of the underlying physical principles.

Related terms

Slope-Intercept Form: A way to write a linear equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Point-Slope Form: A way to write a linear equation in the form $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope.

System of Linear Equations: A set of two or more linear equations with the same variables, which can be solved simultaneously to find the values of the variables that satisfy all the equations.