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The Key Basics of Linear Equations

What You Should Know before Studying Linear Equations

Do you feel bad about algebra homework? You don’t have to because you’re not alone, and there are other students who feel the same way. For example, if you need to study linear equations, keep in mind that this subject is not suitable for many students simply because they prefer easier disciplines and tasks, such as their writing a compare and contrast essay. The main problem is that you need to do your homework that includes endless numbers and alphabets, and this is what makes it quite hard to handle algebraic assignments. If you want to learn how to solve different linear equations, there are certain tricks that will help you ease this task and improve your grades. If they still look complicated and frustrating, take into consideration a great alternative offered by freelancers who can do anything, including your writing an essay introduction, so look for their contact details on the Internet.

For students who study algebra, be prepared for certain equation types because you’ll come across them more often than others. Once you learn how to determine the right type to work with, it becomes much easier to solve given problems because you know the properties of particular linear equations. First, it’s advisable to become familiar with the basic types of algebraic equations.

Linear equations. Remember that their general form is y = mx + b, where b and m are numbers, but the latter one can’t be 0. One of the most effective ways to determine this type is to search for x without any exponents (but it must be the only variable that you have in addition to y). This means that linear equations have no square roots or other exponents, and x is always in a numerator, so it can’t be in a denominator. These equations are called this way because you will end up with a line when graphing them. It’s advisable to take into account that you must see only 1 x and think about a linear that has 1 line. Another useful idea is to build a specific table because it will help you keep all linear equations well-organized.

Quadratic equations. Their general form is ax^2 + bx + c = 0, and this is where c, a, and b are numbers, but a can’t be 0, while others can be 0. The main thing that you should look for is x^2. The highest exponent is the one of 2 so that you shouldn’t be able to see others in such equations. It’s obvious that they are different from linear equations, and that’s why they are grouped in a separate category.

Cubic equations. Their general form is ax^3 + bx^2 + cx + d = 0, as you can guess, a, b, c and d are numbers and they can’t be 0. You should use one effective way to identify them by looking for x^3, and this means that three must be the highest exponent. Besides, if b and c can’t be 0, it’s possible to get x^2 and x terms, but their exponents will never be higher than three, so use this knowledge when doing your relevant case study.

Polynomial equations. As you already know, cubic, quadratic, and linear equations are limit the highest exponent to one, two, and three, this type can take away this limit, and that’s why it’s unique. Pay attention to the word «polynomial» because it means something that consists of a few terms. This is what opens up a number of equations, including cubic, quadratic, and linear equations, but the only restriction is that exponents must be positive whole numbers so that they can’t be negative or fractions.

Basic Facts about linear equations

They are the algebraic equations with every term either the product of a constant or a constant itself and one variable. When it comes to constants, they can be parameters, numbers, and even non-linear functions of specific parameters, but the distinction between parameters and variables usually depends on a given problem.

As a math student, you should understand that linear equations may have 1 and more variables and they occur abundantly in many mathematics subareas, especially in applied mathematics. It’s true that they arise naturally when you model different phenomena, but linear equations are quite useful because many non-linear types can be reduced to them by assuming that the quantities of interest may vary only to some small extent from a background state. Remember that linear equations can’t include any exponents, and they can be categorized as:

  • Equations with 1 unknown x.
  • Linear equations with 2 variables (x and y).
  • The ones that have more than 2 variables.

What about existing forms for their 2-dimensional types? When studying linear equations, you should realize that it’s easy to rewrite them into different forms as you only need to use the right law of elementary algebra. Keep in mind that they are often called equations of a straight line where t, x, y, and θ are variables, while other letters usually represent constants or fixed numbers.

Slope-intercept is the first commonly accepted form of linear equations. To get a better idea of it, you need to be aware of a line, slope, and its y-intercept (this is when a line crosses a y-axis). Their general form is y = m x + b (where m is a slope and b is a y-intercept). If you are aware of the line slope, it’s easy to determine where it will cross a y-intercept or y-axis so that you can write the equations of a line fast. You also need to learn how to take linear equations of other forms to convert them into the slop-intercept ones, and this goal is easy to achieve if you re-arrange them by using equality properties. Another thing that you need to know is how to move easily between different graphical and algebraic representation of specific linear relationships.

The second form of linear equations is standard, and it can be presented as Ax + By = C. You should understand that it’s much harder to get this form directly from graphs, unlike the slope-intercept one, and that’s because C, B, and A are not slopes, intercepts, and other characteristics of a line. One of the most effective methods to write these linear equations from any graphed line is finding a line in other forms, including the slope-intercept one, and re-arranging them into a standard form. The main beauty of this equation type is that it’s easy for you to determine y and x intercepts from it and use given points for graphing. To complete this task successfully, you need to make a specific x-y table. Besides, make sure that you know how to convert these linear equations into their standard form.

The third one is a point-slope form of the line, and it’s represented as y-y1 = m(x-x1). This is where m is a slope, while (x1,y1) is the point on a line. These equations are quite helpful for different reasons, and the main one is that it’s easy to graph a line if you know a point on a slope and a graph. Ensure that you can look at a graph to be able to tell this form of linear equations and learn how to take them in this form to draw a particular graph. Another important academic skill is your ability to convert between this form and others. You should focus on its similarities and a slope of a line.

The System of linear equations

Interested in mathematics? Then you need to understand that this system is the collection of 2 and more linear equations that have the same set of variables. You can say that it’s nothing but 2 and more equations that are solved simultaneously. Nowadays, it’s widely used by businessmen who need to predict future financial events so that they start modeling different real-world situations into the system of equations, as this is what helps them find an effective solution and run their business successfully. This means that you can make quite accurate predictions by using this system of linear equations.

If their graph intersects at a certain point, then your solution will be the ordered pair that corresponds to it, and this means that this system has only 1 solution so that all equations are independent. If their graph coincides, all points on a line can your solution to this system, which has an infinite number of possible solutions (your equations are dependent). Besides, if equation graphs are parallel, then their system has no available solution because parallel lines can’t intersect each other. If your system of linear equations has 1 solution or infinitely many solutions, it’s called consistent. However, if it hasn’t any solution, you can call it inconsistent.

Tips on How to Draw linear equations

If you feel stuck because you don’t know how to complete this academic task without your calculator, keep in mind that drawing their graphs is a simple process. You only need to learn a few basic things about a given equation before getting started and taking the following steps.

First, you should ensure that given linear equations are represented by the formula y = mx + b, which is called a y-intercept form. The good news is that it’s the simplest form that you can use to graph such equations, and take into account that their values don’t have to be whole numbers. As a math student, you can often see the ones that have gradient and slope forms.

The next thing that should be done is plotting b on a y-axis, and it’s always a rational number. You need to find its equivalent on a given y-axis to put a number on this point on a vertical axis if you want to write your term paper on linear equations successfully. Don’t forget to convert m into the fraction (sometimes, the number that you can see in front of x is the necessary fraction so that there is no need to convert it). It’s easy to complete this task because you only need to place the m value over one. The 1st numerator or number is the so-called rise in rise over run, and it’s all about how far a line can travel either vertically or up. The 2nd number or denominator is called the run in rise over run, and it indicated how far a line can travel either horizontally or to the side.

Next, start extending a line from b with the help of slope, and this means that you need to start at the b value because you already know that given linear equations pass through this point. Be sure to extend this line by taking a slope and using its values to come up with the necessary points on equations. Keep extending a line indefinitely by using a standard ruler, while m and a slope will be your guides. Once you complete this process, your required linear equations are graphed.

How to Complete Your Algebra Homework Fast

One of the most common mistakes made by math students is that they always try to memorize everything. You should understand that it won’t help you memorize different algebraic tasks, including linear equations. The best thing you can do is to work toward understanding how these tasks can be easily tackled, as this is how you will learn how to solve algebraic problems.

Be sure to use a variety of interactive study models, and this means that you need to watch more instead of reading all the time. When learning algebra and linear equations, it’s advisable to watch relevant interactive videos, so browse the Internet to find them. There are many online videos that help struggling math students learn how to do their homework, such as dissertation methodology. Give this alternative a try and you will see how it works!

You also need to look for helpful examples because they will help you understand how to start solving specific problems, especially when dealing with completely new topics. Make sure that you understand the questions asked to give correct answers to given algebraic problems. You should understand what they are all about, or you won’t be able to solve linear equations. Finally, familiarize yourself with different patterns to remember as many formulas as you can and pay attention to basic concepts.

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