Linear Equations: _{Relationships with two variables}

What are linear equations?

The formula relating femur length to estimated height in the scenario above is one example of a linear equation—a mathematical statement in which the highest exponent on any variable is 1 (none of the variables are squared, cubed, raised to the fourth power, etc.). They are also known as first-order equations.

As you might expect from the name, when graphed on the Cartesian coordinate system (the familiar x- and y-axis system), a linear equation produces a straight line (Figure 2). Other examples of linear equations include:

This equation converts degrees Celsius (x) to degrees Fahrenheit (y).

An equation that gives the proportion of oxygen atoms (y) to carbon atoms (x) in carbon dioxide.

This equation relates the number of chirps per minute (y) made by a snowy tree cricket to the ambient temperature in degrees Fahrenheit (x).

Early history of linear equations

Linear equations and other basic concepts of algebra have a long history stretching back thousands of years. The ancient Mesopotamians, Egyptians, Greeks, Chinese, and Indians all developed mathematical methods that served as early foundations for modern algebra. But most historians consider the father of algebra to be Abu Abdullah Muhammad ibn Musa al-Khwarizmi (780–850 CE), a scholar with the House of Wisdom academy in what is now Baghdad. In fact, the word algebra comes from al-jabar, a term al-Khwarizmi used to describe the technique of adding equal quantities to both sides of an equation in order to simplify it.

But the math al-Khwarizmi and his predecessors practiced looked very different than what we think of as algebra today. Perhaps the biggest difference is that al-Khwarizmi did not use mathematical symbols. He did not use variables to stand in for unknowns or constants, nor did he use symbols to denote the operations like addition and subtraction that he performed on them. Instead of working with equations, every calculation al-Khwarizmi made was described in words—mostly everyday language with a few technical terms, like al-jabar. He usually wrote about the math needed for practical purposes, such as for dividing inheritance or digging canals.

Today, the use of symbols and equations is so central to algebra that it’s logical to ask: Why are al-Khwarizmi’s word problems even considered algebra? The key features are:

• solving for an unknown quantity (which separates them from simple arithmetic),
• taking a numeric approach (rather than a purely spatial or geometric approach as many Greek scholars did), and
• articulating general rules or techniques for working with numbers (such as al-jabar).

Al-Khwarizmi also studied arithmetic, especially as it was practiced in India. Building upon early Indian scholars, he wrote one of the first known texts describing a decimal system, the operations we now call multiplication and division, and a small circle that appears to be used like a zero (Figure 3).

During the 12th Century, parts of al-Khwarizmi’s writings were translated into Latin and read by scholars working in Europe. These scholars gradually introduced symbols for operations, numbers, and variables. This eventually led to the development of equations as we think of them today.

Development of the Cartesian coordinate system

In the 17th century, another innovation helped connect algebra with geometry. René Descartes, a French philosopher and mathematician, developed a way to visualize equations with two variables by graphing them as lines (linear) or curves (nonlinear). The Cartesian coordinate system, named for Descartes, is a system of two perpendicular axes, usually labeled x and y. It is an important tool in modern math from algebra to calculus, and scientists frequently use it to visualize the relationship between two variables in their data. (For more about how scientists use the Cartesian coordinate system, see our module Using Graphs and Visual Data in Science.)

Descartes’ 1637 book La géométrie described the basic idea of a coordinate system as well as an organized collection of symbols and conventions we still use today. From La géométrie, we get things like the modern square root symbol (radicand) and the convention of writing an exponent as a small, raised numeral just after its base. The practice of using lower-case letters from the beginning of the alphabet to stand for given numbers or constants and using lower-case letters from the end of the alphabet to represent variables also comes from Descartes. Today we call the equation ax + by = c the standard form of a linear equation.

Working with linear equations

When presented with a linear equation, if we know the value of one of the variables (often x or y), we can solve the equation for the other variable. Let’s use the cricket equation shown in Figure 2 (Dolbear, 1897) as an example. If we sit outside one evening and count the snowy tree crickets chirping 80 times per minute (y = 80), we can solve for the temperature (x) as follows:

Plug in 80 chirps for y.

Simplified.

Thus, we know the temperature is approximately 60° F. The following evening if we observe the crickets chirping 60 times per minute (y = 60), we can plug that value in and calculate the new value for x (55° F).

In the cricket equation, x is known as the independent variable, and y is the dependent variable, since its value depends on the value of x (chirping rate depends on temperature). Sets of x and y values that make the equation true are solutions to the equation. They are generally written as ordered pairs in the form (x, y). One solution to the cricket equation is the ordered pair 60° F and 80 chirps per minute (60, 80). In theory, there is an infinite number of solutions to the equation, including (55, 60), (65, 100), and (43, 12). Each ordered pair is a point on the line described by the equation (Figure 4).

However, some ordered pairs that are solutions to a given equation may not make sense in the real world. For example, the ordered pairs (37, -12) and (200, 640) are valid solutions to the cricket equation but don’t make sense in this context. Crickets cannot chirp -12 times per minute, and they are unlikely to be alive and chirping (let alone at a pace of 640 chirps per minute) if the temperature has reached 200° F. When working with an equation in a real-world scientific context, it is important to reflect on a given solution and consider whether it makes sense in that context.

If we don’t know the value of either variable, we can still solve for one of the variables in terms of the other.

Again using the equation above, we can solve for y in terms of x.

Subtract 40 from both sides.

Multiply both sides by 4.

Simplify both sides of the equations.

Rearrange the equation so y is on the left.

Thus, we can see that y equals four times x, minus 160. If we are out in the field and already know the temperature (x), this equation can be used to quickly calculate the expected number of chirps (y).

Slope-intercept form

Additionally, if we want to visualize a linear equation by graphing it, the arrangement shown above, called slope-intercept form, is often more useful. Slope-intercept form follows the general format:

Where x and y are the variables and m and b are constants. (Keep in mind that the m and b may be positive, negative, or equal to zero.)

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.

The slope is a ratio relating the change in y to the change in x (sometimes called “rise to run”) from one point to another on a line. For example, if m = ½, each point on the line is 1 unit higher on the vertical (y) axis for every two units to the right on the horizontal (x) axis. If m = -3, each point on the line is 3 units lower on the y-axis for every one unit to the right. 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. Imagine a study where a scientist measures an organism’s length as it grows over the course of a month. If he finds the growth rate is linear and writes an equation in the form y = mx + b to describe the organism’s length, b would likely indicate the organism’s length at the beginning of the study.

Describing vertical and horizontal lines with linear equations

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. In the equation above relating femur length (L) to a person’s overall height (H), the taller the person is, the longer his or her femur. But, if we consider the relationship between height (H) and number of limbs (N), we see no dependence of one upon the other. Regardless of height, N = 4 describes number of limbs in all cases.

A vertical line has an undefined slope and thus cannot be written in slope-intercept form. The general equation for a vertical line is x = a, where a is a constant. On a vertical line, all points have the same x-value, and the line never crosses the y-axis (unless the equation is x = 0). A horizontal line has a slope of 0, so the slope-intercept form can be simplified to y = b, where b is the y-intercept, as well as the y-value in every ordered pair that satisfies the equation (Figure 6).

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.)

Using linear equations 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.

How to calculate a rate

A rate is a measurement of change relative to time. Scientists often need to know how quickly or slowly (“at what rate”) a given process is occurring. A geologist, for example, may want to know the rate at which pieces of the Earth’s crust are moving in order to assess potential seismic hazards. 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 (t₁ and t₂) and two corresponding values for the condition that is changing (d₁ and d₂). So for example:

where d₂ is the distance traveled at time t₂ and d₁ is the distance traveled at time t₁. The Greek letter Δ, “delta,” means change, and you will often see it used in rate calculation problems. Written using delta, our example rate equation becomes:

The rate equals the change in distance (d) over the change in time (t). Let’s look at a real-world example.

Sample Problem 1

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 1928, the storage capacity of the reservoir was 300,000 acre-feet. In 1993, the USGS determined that the buildup of sediment had reduced the reservoir’s capacity to 189,000 acre-feet (Langland & Hainly, 1997). Assuming the rate of sediment deposition has been constant over that time period, at what rate (in acre-feet per year) is the reservoir’s storage capacity being reduced?

Solution 1

In this example, the changing condition (c) is the reservoir’s capacity in acre-feet. And time (t) is measured in years:

The reservoir is losing storage capacity at an average of 1,708 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.

Sample Problem 2

In 2006, scientists working with the Plate Boundary Observatory Network began closely tracking the location of a GPS station west of the San Andreas Fault in California, meaning that it is on the Pacific Plate. The station is moving slowly northwest as the Pacific Plate and the North America Plate grind past one another. In May 2007, researchers recorded the station 32.95 mm northwest of its original position. In May 2012, they recorded it 195.30 mm northwest of its original position (UNAVCO, 2012). On average, how fast was the station (and thus the Pacific Plate) moving between 2007 and 2012?

Solution 2

To solve for the station’s average rate of movement we need to know how far it was displaced between 2007 and 2012 (Δx).

Since the time period of interest is between 2007 and 2012, we know that \(\Delta t = 5\) years. Therefore:

Between 2007 and 2012, the plate moved an average of approximately 32.5 mm northwest per year.

Sample Problem 3

If the plate continues moving at the same rate in the same direction, how far will it be from its original (May 2006) position by May of 2050?

Solution 3

Now that you have calculated the plate’s rate of movement, you can calculate how far it will travel using the general equation:

Where d stands for distance, r stands for rate of movement, and t stands for time elapsed.

By May 2050, the station will have moved approximately 1,430 millimeters.

How to convert units of measurement

When making a calculation in science (or daily life), it’s important to make sure you use consistent units of measurement—and accurately convert from one unit to another when needed. The metric system (also called SI, for Système Internationale) is used by scientists all around the world and by most countries. But if you live in or travel to the United States or a handful of Caribbean countries, you may need to convert between SI units and English units. Regardless of where you live, being able to convert between units comes in handy for finding the product that is the best deal per unit weight at the market, converting currency, and converting between different types of SI units in science. For more information, see our The Metric System: Metric and Scientific Notation.

Sample Problem 4

Your friend works for a US newspaper and wants to report on the findings of the scientists in Sample Problem 2 above. Knowing that the paper’s audience is more familiar with inches than millimeters, he wants to convert the rate of plate movement from millimeters per year to inches per year and asks for your help. What rate should you tell your friend to report?

Solution 4

To make this conversion, you need to know how many millimeters there are in an inch. You look up the follow conversion factor:

Now you can convert the plate’s rate of movement to inches per year:

Your friend should report that the plate is moving at an average rate of approximately 1.28 inches per year.

For more on converting units, including a description of the factor-label method for solving equations, see our module Unit Conversion: Dimensional Analysis.

Non-linear relationships

There are many relationships in science that cannot be described by a linear equation. For example, in biology, tissue growth in some species occurs at different rates throughout an organism’s life and cannot be described by a single equation. (Think of how quickly a baby grows compared to a teenager or an adult.) In Earth science, lava flows often occur in spurts as a volcano goes through active and quiet periods. In these cases, it doesn’t necessarily make sense to describe growth or flow rates with a single equation or to calculate an average annual rate. Other phenomena, such as growth of a population, cell division, or the rate of some chemical reactions, may occur at exponential rates. These relationships are expressed in exponential equations, which produce Cartesian graphs with curved lines instead of straight lines.

Linear equations are an important tool in science and many everyday applications. They allow scientist to describe relationships between two variables in the physical world, make predictions, calculate rates, and make conversions, among other things. Graphing linear equations helps make trends visible. Gaining a strong understanding of linear equations both helps in scientific problem solving and lays a foundation for exploring other, more mathematically complex relationships in science.