Linear Equation

Definition

A linear equation is a mathematical equation in which the dependent variable is expressed as a linear function of one or more independent variables. This means the equation can be represented as a straight line when graphed on a coordinate plane.

5 Must Know Facts For Your Next Test

1. The graph of a linear equation is always a straight line, which can be described by its slope and $y$-intercept.
2. Linear equations can be used to model and analyze real-world situations, such as the relationship between time and distance traveled.
3. The slope of a linear equation represents the rate of change between the dependent and independent variables.
4. Linear equations can be solved using various methods, including graphing, substitution, and elimination.
5. The solutions to a system of linear equations can be found using techniques such as the elimination method or the substitution method.

Review Questions

• Explain how the slope-intercept form of a linear equation, $y = mx + b$, can be used to describe the characteristics of a line.
  • The slope-intercept form of a linear equation, $y = mx + b$, provides valuable information about the characteristics of the line. The slope, $m$, represents the rate of change between the $x$ and $y$ variables, indicating how the $y$-value changes for a unit change in $x$. The $y$-intercept, $b$, represents the point where the line crosses the $y$-axis, giving the starting value of the line. By understanding the slope and $y$-intercept, you can determine the direction, steepness, and starting point of the line, which is crucial for analyzing and interpreting the relationship between the variables in a linear equation.
• Describe how the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, can be used to find the equation of a line given a point and the slope.
  • The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, allows you to find the equation of a line when you know a point $(x_1, y_1)$ that lies on the line and the slope, $m$, of the line. By rearranging the equation, you can solve for the $y$-intercept, $b$, to obtain the slope-intercept form, $y = mx + b$. This is particularly useful when you need to find the equation of a line given a specific point and the rate of change (slope) between the variables. The point-slope form provides a direct way to incorporate the known information about the line into the equation, making it easier to determine the complete linear equation.
• Explain how the standard form of a linear equation, $Ax + By = C$, can be used to analyze and solve systems of linear equations.
  • The standard form of a linear equation, $Ax + By = C$, is particularly useful when working with systems of linear equations. In a system of linear equations, multiple equations with different coefficients and constants are presented, and the goal is to find the values of the variables that satisfy all the equations simultaneously. By expressing the equations in standard form, you can more easily apply techniques like the elimination method or the substitution method to solve for the unknown variables. The standard form also allows you to analyze the relationships between the equations, such as determining whether the system has a unique solution, no solution, or infinitely many solutions. Understanding the standard form of linear equations is crucial for effectively solving and interpreting systems of linear equations, which are commonly encountered in various mathematical and scientific applications.

Related terms

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

Point-Slope Form: The form of a linear equation written as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope.

Standard Form: The form of a linear equation written as $Ax + By = C$, where $A$, $B$, and $C$ are constants.