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How are linear equations used?
Linear equations are handy whenever one quantity changes directly as another one changes. For instance, the amount of gas you need to drive a certain distance, or how far something might travel at any particular speed. The equation can give you the answer without having to draw a graph.
It can also tell you where multiple lines on the same graph would cross, without actually drawing the lines. Linear equations are an example of algebra influencing geometry. While traditional geometry meant you had to draw diagrams with a ruler and compass, using just the algebraic formulas means you can find the same answer by manipulating equations.
Who invented linear equations?
Linear equations are as old as algebra, although their use in graphing relationships (like where two lines cross) falls under coordinate graphing, which is an idea credited to Descartes, in the 17th century. The main ideas were expanded into more than two dimensions and multiple variables with the advent of linear algebra, which is really a product of the 20th century and many mathematicians.
What are linear equations?
The word linear means 'in a row' or 'in a line'. So, a linear equation is a mathematical way to describe a straight line. (A curved line is called curvilinear.) Linear equations describe the line you would see on a graph when the line is perfectly straight.
Here is a linear equation: y = 2x + 1
The same line looks like this on a graph:
The arrows on the ends of the line mean it continues past the part of the graph we have shown here. The important thing is that the equation completely describes this line. To check this, we can see what value Y should be when X is 2. Our equation is Y = 2X +1, and we can replace X with the number we want to check: Y = 2(2) + 1 = 4 + 1 = 5.
Take a look at the graph and you will see that when X = 2, Y does equal 5. The advantage of the equation is that it will give us the right answer no matter what X is, even something like 7892. (Imagine how big the graph would have to be to show such a huge number!)
A basic problem with linear equations.
One problem is determining where the line will cross the X or Y axis. In our example, where would the line cross the X axis? Even though it's not drawn above, we can determine where by setting Y = 0 (because Y is zero everywhere on the X axis). With Y at zero, our linear equation looks like this: 0 = 2X + 1.
2X must be equal to -1, so that we'll get zero on the right side (0 = -1 + 1). If 2X equals -1, then X must be -1/2. So, the line will cross the X axis at -1/2.
An interesting fact about linear equations.
In the general formula, the number in front of the X is called the slope. It tells you how steep the line will be on a graph. Our example used the formula Y = 2X + 1 and the 2 in front of the X means the slope for that line is 2.
The bigger the number, the steeper the line will appear on a graph. And a line that goes straight up has a slope of infinity!