Linear Equation

Definition

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

5 Must Know Facts For Your Next Test

1. Linear equations can be solved using the addition and subtraction properties of equality, which involve isolating the variable on one side of the equation.
2. Equations with variables and constants on both sides can be solved by first combining like terms and then using the addition and subtraction properties of equality.
3. Formulas, which are linear equations, can be solved for a specific variable by isolating that variable on one side of the equation.
4. The rectangular coordinate system is used to graph linear equations, with the $x$-axis representing the horizontal axis and the $y$-axis representing the vertical axis.
5. The slope of a line, which represents the rate of change between two points, is a key characteristic of a linear equation and can be used to graph the equation.

Review Questions

• Explain how the addition and subtraction properties of equality are used to solve linear equations.
  • The addition and subtraction properties of equality state that the same quantity can be added to or subtracted from both sides of an equation without changing its value. This allows you to isolate the variable on one side of the equation by performing the same operations on both sides. For example, to solve the equation $2x + 5 = 11$, you would first subtract 5 from both sides to get $2x = 6$, and then divide both sides by 2 to find that $x = 3$. This step-by-step process of using the addition and subtraction properties is a fundamental technique for solving linear equations.
• Describe how the rectangular coordinate system is used to graph linear equations.
  • The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane with a horizontal $x$-axis and a vertical $y$-axis. Linear equations can be graphed on this coordinate plane by plotting points that satisfy the equation. The $x$-intercept, where the line crosses the $x$-axis, and the $y$-intercept, where the line crosses the $y$-axis, are important features that can be used to graph the line. Additionally, the slope of the line, which represents the rate of change between any two points on the line, can be used to determine the orientation and steepness of the line.
• Analyze how the concept of slope is used to understand the properties of a linear equation.
  • The slope of a linear equation is a fundamental characteristic that describes the steepness and direction of the line. Slope is defined as the change in the $y$-value divided by the change in the $x$-value between any two points on the line. This ratio represents the rate of change between those points. The slope of a line can be used to determine important properties, such as the line's orientation (positive, negative, zero, or undefined), the line's steepness, and the relationship between the variables in the equation. Understanding the slope of a linear equation is crucial for graphing the line, interpreting its meaning, and making connections between the algebraic representation and the geometric representation of the equation.

Related terms

Variable:

A symbol, usually a letter, that represents an unknown or changing quantity in a mathematical expression.

Coefficient:

The numerical factor multiplied by a variable in an algebraic term.

Constant:

A fixed value that does not change in a given context or problem.