key term - Linear equation

Definition

A linear equation is a mathematical statement that establishes a relationship between variables, represented in the form of an equation where each term is either a constant or the product of a constant and a single variable. This type of equation forms a straight line when graphed on a coordinate plane, which is a fundamental concept in understanding algebraic operations. Linear equations are often used to express real-world situations where there is a constant rate of change, making them essential in various fields such as economics, physics, and engineering.

5 Must Know Facts For Your Next Test

1. Linear equations can be expressed in several forms, including slope-intercept form ($$y = mx + b$$), standard form ($$Ax + By = C$$), and point-slope form ($$y - y_1 = m(x - x_1)$$).
2. The graph of a linear equation is always a straight line, and the degree of the equation is one, indicating that it is a first-degree polynomial.
3. Linear equations can have one solution (intersecting lines), infinitely many solutions (coincident lines), or no solution (parallel lines).
4. Operations such as addition, subtraction, multiplication, and division can be applied to both sides of a linear equation without changing its solution.
5. The concept of linearity also extends to systems of linear equations, which involve multiple equations that can be solved simultaneously to find common solutions.

Review Questions

• How can you determine if two linear equations represent parallel lines?
  • Two linear equations represent parallel lines if they have the same slope but different y-intercepts. This means that when written in slope-intercept form ($$y = mx + b$$), both equations will have identical values for 'm' but different values for 'b'. As a result, these lines will never intersect on a graph.
• Describe how you would solve a system of linear equations using substitution.
  • To solve a system of linear equations using substitution, start by solving one of the equations for one variable in terms of the other. Next, substitute this expression into the other equation. This creates a new equation with one variable that can be solved. After finding the value for one variable, substitute back into one of the original equations to find the corresponding value for the other variable, resulting in an ordered pair that represents the intersection point.
• Evaluate how changes in the coefficients of a linear equation affect its graph.
  • Changing the coefficients in a linear equation affects both its slope and y-intercept, which in turn alters its graph. For instance, increasing the coefficient of 'x' (the slope) will make the line steeper, while changing the constant term will shift the line up or down on the y-axis. Understanding these transformations is essential for analyzing how modifications to a linear model reflect changes in real-world scenarios.

Related terms

slope:

The slope is a measure of the steepness or incline of a line, calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line.

y-intercept:

The y-intercept is the point where a line crosses the y-axis, representing the value of y when x is equal to zero in a linear equation.

solution: A solution to a linear equation is any ordered pair (x, y) that satisfies the equation, meaning when substituted back into the equation, both sides remain equal.