Who invented system of equations

In this form, m is the slope of the line — a ratio that tells us how much a line rises over a given distance. Meanwhile, b is the y-intercept — the point where the line crosses the y-axis.

Keep in mind that in real-world contexts, the axes may be labeled with variables other than x and y. For example, you may see t to represent time or v to represent velocity. For this reason, it may help to think of the y-intercept as the vertical-axis-intercept. A positive slope means that a line trends up as one moves to the right; a negative slope occurs when a line trends down as one moves to the right Figure 5.

For linear equations used in science, b often represents the starting point. Vertical and horizontal lines are also described by linear equations. In a scientific context, a horizontal or vertical line indicates that a variable is constant , regardless of changes in any other variable. But, if we consider the relationship between height H and number of limbs N , we see no dependence of one upon the other.

A vertical line has an undefined slope and thus cannot be written in slope-intercept form. In real world applications such as those described below, each axis—and thus each variable—represents a measurement of some factor, such as distance traveled, time elapsed, degrees Fahrenheit, etc.

The linear equation describes the relationship between the two measurements. Although x and y are the default variables for the axes, you will often see other letters used in equations and on graphs that hint at what the variable represents. For example, t may be used for time, d for distance, etc. See our module Using Graphs and Visual Data in Science for more about how graphs are used in science.

Linear equations can be used to describe many relationships and processes in the physical world, and thus play a big role in science. Frequently, linear equations are used to calculate rates, such as how quickly a projectile is moving or a chemical reaction is proceeding. They can also be used to convert from one unit of measurement to another, such as meters to miles or degrees Celsius to degrees Fahrenheit. In some cases, scientists discover linear relationships during the course of research.

For example, an environmental scientist analyzing data she has collected about the concentration of a certain pollutant in a lake may notice that the pollutant degrades at a constant rate.

Using those data, she may develop a linear equation that describes the concentration of the pollutant over time.

The equation can then be used to calculate how much of the pollutant will be present in five years or how long it will take for the pollutant to degrade entirely. A rate is a measurement of change relative to time. A chemist may need to know the rate at which two substances react with one another in order to understand the products of a chemical reaction. A rate r is calculated by determining the amount of change for example, distance traveled and the time elapsed. To do this, we need two values for time t 1 and t 2 and two corresponding values for the condition that is changing d 1 and d 2.

So for example:. Written using delta , our example rate equation becomes:. The rate equals the change in distance d over the change in time t.

When the Susquehanna River reaches the Conowingo Reservoir in Maryland, the water flow slows, and much of the sediment the river has carried downstream settles out behind the Conowingo Dam.

When the dam was originally built in , the storage capacity of the reservoir was , acre-feet. And time t is measured in years:. The reservoir is losing storage capacity at an average of 1, acre-feet per year. Thus the rate of change in capacity appears as a negative value. You can also visualize a rate by graphing it Figure 7. The greater the slope of the line, the faster the rate.

By looking at the graph, you can see approximately how much storage capacity the reservoir lost after one year, 10 years, or any other length of time. And perhaps more importantly, you can predict when the reservoir will lose all of its capacity and need to be dredged or removed. The station is moving slowly northwest as the Pacific Plate and the North America Plate grind past one another.

In May , researchers recorded the station In May , they recorded it On average , how fast was the station and thus the Pacific Plate moving between and ? Cardan , in Ars Magna , gives a rule for solving a system of two linear equations which he calls regula de modo and which [ 7 ] calls mother of rules!

Cardan therefore does not reach the definition of a determinant but, with the advantage of hindsight, we can see that his method does lead to the definition. Many standard results of elementary matrix theory first appeared long before matrices were the object of mathematical investigation. This amounts to diagonalising a symmetric matrix but de Witt never thought in these terms.

The idea of a determinant appeared in Japan before it appearedin Europe. In Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables in exactly the way the Chinese methods described above were constructed. Without having any word which corresponds to 'determinant' Seki still introduced determinants and gave general methods for calculating them based on examples.

The first appearance of a determinant in Europe was ten years later. References show. A Jennings, Matrices, ancient and modern, Bull. Muhammad ibn Musa al-Khwarizmi wasn't the only Arab mathematician to do important work relating to algebra at this time.

He wrote books and included information on how to solve equations to the third degree. This took his work further than any of the other Arabic mathematicians that had come before him. His method of solving cubic equations is very important and is used today. In the 16th century, there were a few developments in algebra that occurred in Europe.

First of all, Michael Stifel wrote a comprehensive study on arithmetic and algebra. It was the first algebraic text to be written in the German language. Johannes Widmann also gave important lectures on algebra and its uses at that time in the 16th century.

Both men were influential in bringing algebra to the fore in Europe during this particular period of history. Most historians of mathematics now agree that it was Rene Descartes who was responsible for this particular development. It's a symbol that is still used by mathematicians and students. Descartes began by using a range or letters before eventually selecting x as the most common variable name.

It's also important to mention Gottfried Leibniz. He was the first to explicitly employ the notion of a function as a relation between a set of mathematical inputs and outputs. This was very important to algebra at the time, because it allowed functions to be written that described physical processes in the real world. He was also responsible for discovering Boolean algebra, as well as symbolic logic. So, although Leibniz appears quite late in the historical development of algebra, he had a big impact on it.

In summary, many different cultures and people developed algebraic theory. And each breakthrough and new method came about for its own particular reasons. It was always done to solve a problem and make a solution easier to find. For example, the Babylonians used algebra to work out the area of items and the interest on loans, among other things. It had a real use and purpose and this why it was developed.

The Hellenistic Greek mathematician Diophantus used algebra for similar reasons, but he was much more interested in exact solutions than the Babylonians, who tended to use approximations. In the centuries since ancient Greeks and Babylonians, we have used algebra to solve a great many problems in a wide variety of subjects in science and engineering.

Al-Khwarizmi was focused on solving computations problems, and his work has been revisited in recent decades. His work also helped solve trade and inheritance problems. Today, algebra is used extensively in engineering and construction planning to ensure that buildings, bridges, airplanes, and more are built safely and correctly. In the financial sector, algebra is used in predicting risks and in assessing economic impacts.