key term - Linear Equation

Definition

A linear equation is a mathematical equation that represents a straight line on a coordinate plane. It is characterized by a constant rate of change, or slope, and a starting point, or y-intercept, that together define the line's position and orientation.

5 Must Know Facts For Your Next Test

1. Linear equations can be used to solve for unknown variables when there are an equal number of equations and unknowns.
2. The general form of a linear equation is $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.
3. Graphing linear equations on a coordinate plane allows for the visualization of the relationship between the variables.
4. The slope-intercept form of a linear equation, $y = mx + b$, is useful for determining the line's direction and starting point.
5. Systems of linear equations can be solved using various methods, such as graphing, substitution, or elimination, to find the values of the unknowns.

Review Questions

• How can linear equations be used to solve for unknown variables in the context of 2.3 Solve Equations with Variables and Constants on Both Sides and 2.4 Use a General Strategy to Solve Linear Equations?
  • Linear equations can be used to solve for unknown variables when there are an equal number of equations and unknowns. In the context of 2.3 and 2.4, linear equations can be used to isolate the variable on one side of the equation and solve for its value. This involves performing algebraic operations, such as combining like terms and using the addition or multiplication property of equality, to manipulate the equation and find the solution.
• Explain how the concepts of the rectangular coordinate system (4.1) and graphing linear equations in two variables (4.2) are related to the understanding of linear equations.
  • The rectangular coordinate system provides a visual representation of the relationship between two variables, which is essential for understanding and graphing linear equations. By plotting points on the coordinate plane, the linear equation can be represented as a straight line, allowing for the visualization of the relationship between the variables. This, in turn, helps in understanding the properties of linear equations, such as slope and y-intercept, which are crucial for solving and interpreting linear equations.
• Discuss how the slope-intercept form of a linear equation (4.5) and the ability to find the equation of a line (4.6) are important in the context of solving systems of linear equations by graphing (5.1).
  • The slope-intercept form of a linear equation, $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept, is essential for solving systems of linear equations by graphing. By understanding the slope-intercept form, students can graph individual linear equations and determine their points of intersection, which represent the solution to the system of equations. Additionally, the ability to find the equation of a line, given information such as the slope and a point on the line, allows for the construction of the necessary equations to solve a system of linear equations by graphing.

Related terms

Slope:

The measure of the steepness or incline of a line, represented by the ratio of the change in y-values to the change in x-values between two points on the line.

Y-Intercept: The point where a line intersects the y-axis, representing the value of y when x is equal to 0.

Coordinate Plane:

A two-dimensional plane with perpendicular x and y axes used to graph and represent the relationships between variables.