Linear Equation Facts

A linear equation is a mathematical way of describing a straight line. It has only one or two variables, and neither variable is raised to a power greater than one. In other words, the equation has no squares, cubes or other exponents. Also, the variables are never used as denominators of fractions.

A linear equation describes a line by describing what its slope is and where it crosses the x and y axes on a grid.

Variables

In math, a variable represents a quantity that may change. In that way, it contrasts with a constant, which never changes. For example, pi never changes. It is always equal to a number very close to 3.14159265. But a variable, often represented by an x or a y, can change as the terms in an equation change.

In a linear equation with two variables, one variable is often designated as independent, and one as dependent. The dependent variable changes to match manipulations of the independent variable. The independent variable is represented by an x in a linear equation, and the dependent variable is represented by a y.

Equations

Equations are mathematical sentences that set one thing equal to another. They contrast with inequalities, which say one thing is greater than or less than another. This is an equation: 2 + 2 = 4. This is an inequality: 2+3 > 4.

Linear equations

One form of linear equation directly describes how x is related to y, for example: 2x = y. The x is the independent variable. When we change it, the y will change accordingly. If we set x as 1, then 2(1) = 2. In other words, when x is 1, y is 2. If x is 0, then y is also 0. If x is 2, then y is 4, and so on. The y is always twice as large, except when x is zero, which, of course, is always zero when multiplied by any number.

Relationships

Linear equations describe real relationships, like the relationship between how fast something goes and how far it gets in a period of time. Another linear relationship is the one between Fahrenheit and Celsius temperatures.

Graphing the line

A line is defined by two points on a grid. Therefore, anyone who knows two points that are on a line can graph it. One way to do that is to substitute a number for x, and solve the equation to learn what y is. Do this twice to get two pairs that are the location of two points on the grid. Connect them with a straight line to see the line that the equation describes.

In our example, when x = 2 then y = 4, and when x = 0 then y = 0. Therefore, two points on the graph are (2, 4) and (0, 0). The x is always listed first in the parentheses.

To make a graph, get a piece of graph paper and draw two lines that divide it into quarters. The horizontal line is the x-axis and the vertical line is the y-axis. The point where the two lines cross is (0, 0).

One of the points on the line is at (0, 0), so it is easy to find. To find the other, count two squares to the right from the intersection on the x-axis. Then count four squares up from there. That point is (2, 4). Connect the two points with a straight line, and extend the line. That is the graph of the linear equation 2x = y.

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