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. Almost any situation where there is an unknown quantity can be represented by a linear equation, like figuring out income over time, calculating mileage rates, or predicting profit.
Many people use linear equations every day, even if they do the calculations in their head without drawing a line graph. Linear equations ones that graph as straight lines are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.
There are three possible outcomes for a system of linear equations: one unique solution, infinitely many solutions, and no solution. A system of linear equations is just a set of two or more linear equations. In two variables x and y , the graph of a system of two equations is a pair of lines in the plane. The lines intersect at infinitely many points. The two equations represent the same line. As a rule of thumb, a system is linear, if the operations on the input signal are all linear and no signal-independent terms are contained.
In other words, we have the following equation. Note that since we are dealing with money we rounded the answer down to two decimal places. This problem is pretty much the opposite of the previous example. Therefore, the equation and solution is,. These are some of the standard problems that most people think about when they think about Algebra word problems. The standard formula that we will be using here is. Now, we need to sketch a figure for this one.
From this figure we can see that the Distance Car A travels plus the Distance Car B travels must equal the total distance separating the two cars, miles. We used the standard formula here twice, once for each car. We know that the distance a car travels is the rate of the car times the time traveled by the car. Plugging these into the word equation and solving gives us,. For this problem we are going to need to be careful with the time traveled by each car.
The only difference is what we substitute for the time traveled for the faster car. In this case the slower car will travel for 3. Here is the figure for this situation. Plugging these into the work equation and solving gives,. The standard equation that will be needed for these problems is,. The word equation for this problem is,. We can use the following equation to get these rates. Plugging these quantities into the main equation above gives the following equation that we need to solve.
So, it looks like it will take the two machines, working together, 1. The basic word equation for this problem is,. This time we know that the time spent working together is 3. We now need to find the work rates for each person. We are going to be looking at mixing solutions of different percentages to get a new percentage. The solution will consist of a secondary liquid mixed in with water.
The secondary liquid can be alcohol or acid for instance. Note as well that the percentage needs to be a decimal.
Linear equations use one or more variables where one variable is dependent on the other. Almost any situation where there is an unknown quantity can be represented by a linear equation, like figuring out income over time, calculating mileage rates, or predicting profit.
Many people use linear equations every day, even if they do the calculations in their head without drawing a line graph. Imagine that you are taking a taxi while on vacation. Without knowing how many miles it will be to each destination, you can set up a linear equation that can be used to find the cost of any taxi trip you take on your trip. Linear equations can be a useful tool for comparing rates of pay. 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 ? Between and , the plate moved an average of approximately If the plate continues moving at the same rate in the same direction, how far will it be from its original May position by May of ?
Where d stands for distance, r stands for rate of movement, and t stands for time elapsed. 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. Your friend works for a US newspaper and wants to report on the findings of the scientists in Sample Problem 2 above.
What rate should you tell your friend to report? To make this conversion, you need to know how many millimeters there are in an inch.
You look up the follow conversion factor:. Your friend should report that the plate is moving at an average rate of approximately 1. For more on converting units , including a description of the factor-label method for solving equations, see our module Unit Conversion: Dimensional Analysis.
There are many relationships in science that cannot be described by a linear 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. 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.
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