Linear Equation

Definition

A linear equation is a mathematical equation in which the variables are raised to the first power and the variables are connected by addition, subtraction, or scalar multiplication. These equations represent 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.
2. The slope of a linear equation represents the rate of change between the variables.
3. Linear equations can be used to model and analyze real-world situations, such as the relationship between distance, rate, and time.
4. The $y$-intercept of a linear equation represents the value of the $y$-variable when the $x$-variable is zero.
5. Linear equations can be classified as either homogeneous or non-homogeneous, depending on the presence of a constant term.

Review Questions

• Explain how the slope of a linear equation can be interpreted in the context of a real-world situation.
  • The slope of a linear equation represents the rate of change between the variables. In the context of a real-world situation, the slope can be interpreted as the rate at which one variable changes in relation to the other. For example, in the equation $y = 2x + 3$, the slope of 2 indicates that for every 1 unit increase in the $x$-variable, the $y$-variable increases by 2 units. This could represent the relationship between distance and time, where the slope would be the speed or rate of change.
• Describe the differences between the slope-intercept form, point-slope form, and standard form of a linear equation, and explain when each form might be more useful.
  • The three main forms of a linear equation are the slope-intercept form ($y = mx + b$), the point-slope form ($y - y_1 = m(x - x_1)$), and the standard form ($Ax + By = C$). The slope-intercept form directly shows the slope ($m$) and $y$-intercept ($b$) of the line, making it useful for graphing and interpreting the line's characteristics. The point-slope form is helpful when you know a point on the line and the slope, and the standard form is useful when the coefficients of $x$ and $y$ are not easily expressed as a ratio. The choice of form depends on the information given and the specific needs of the problem or application.
• Analyze how the graph of a linear equation can be used to determine the solution(s) to a system of linear equations.
  • The graph of a linear equation can be used to determine the solution(s) to a system of linear equations. When two or more linear equations are graphed on the same coordinate plane, the point(s) where the lines intersect represent the solution(s) to the system. The $x$- and $y$-coordinates of the intersection point(s) satisfy all the equations in the system simultaneously, providing the value(s) of the variables that make the system true. By analyzing the graphs of the linear equations, you can identify the number of solutions (one, infinite, or no solution) and the specific values of the variables that solve the system.

Related terms

Slope-Intercept Form:

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

Point-Slope Form:

The equation of a linear function written 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.

Standard Form:

The equation of a linear function written in the form $Ax + By = C$, where $A$, $B$, and $C$ are real numbers and $A$ and $B$ are not both zero.