Find All Solutions to the System of Linear Equations Calculator
In mathematics, an equation is a statement that demonstrates the equality of two expressions. These two expressions are separated only by an equals sign (=). They can consist of any number of values and operations, with the only requirement being the expressions have to be equal.
We usually say an equation has two sides: the right-hand side, and the left-hand side. The right-hand side is generally assumed to be 0. However, this doesn't always have to be true (we will see some examples later).
Equations are made up of three parts. Those parts are:
- The constants
- The coefficients
- The variables
The given numbers that stand alone are called the constants. They are called that because they don't change. This means that in any version of a certain equation, they always remain the same.
The coefficients stand with the variables, and they serve to amplify them.
The variables are the unknowns of an equation, the missing parts if you will.
What is a linear equation?
Sir William Rowan Hamilton invented the linear equation in 1843. They are a part of linear algebra, which is a very important branch, not just in mathematics, but other sciences, such as engineering, computing, and many more. The standard formula for one-variable linear equations is:
\underbrace{a}_{\text{{\color{Green}coefficient}}} \cdot \underbrace{x}_{\text{{\color{Red}unkown}}} + \underbrace{b}_{\text{{\color{Blue}constant}}} = 0
They are equations of thefirst degree. This means that all the numbers in the equation must have theexponent1. So, if any of the numbers had an exponent larger or smaller than 1, the equation would no longer be linear, for example:
5 \cdot x^2 + 2 = 10
This equation would now be classified as aquadratic equation. If you wish to learn more about those, you can check out our quadratic equation calculator.
They can also come in two-variable forms:
{\color{Green}a} \cdot {\color{Red}x} + {\color{Green}b} \cdot {\color{Red}y} + {\color{Blue}c} = 0
Linear equation formula and step by step solution
Solving a linear equation means finding the value of x that makes the equality of the two expressions true. When it comes to the one-variable forms, there is a very simple formula for calculating x:
\large \fcolorbox{Red}{Cyan}{$x = - \frac {b}{a}$}
Let's do a quick example:
5 \cdot x + 10 = 0 \\ \boxed {{\color{Red}x} = - \frac {10}{5} = {\color{Red}-2}} \\ 5 \cdot {\color{Red}-2} + 10 = 0
This is if we assumeais not 0. Ifa is 0, there are two possible outcomes. In the case thatbis also 0, x can be any number, and the equality will be true. Ifb is not 0 however, there is no solution to the equation, and it is dubbed inconsistent.
With this formula, you can even use a standard algebra calculator to solve the equation.
How to use the linear equation calculator
Our calculator is pretty simple and easy-to-use. You can use it to solve every one-variable linear equation. All you have to do is enter the coefficient and the constant, and our tool will calculate the unknown. Let's use the example from before:
5 \cdot x + 10 = 0
So, in our calculator you would enter:
And the calculator would get:
But what if the equation you have doesn't have 0 as the end result? Well, there is a simple solution for that too. All you have to do is transfer the result to the left side of the equation, and 0 will be left on the right side. Let's do a quick example:
10x+10=5
Now, if you put this into our calculator, you wouldn't get the right result, because this is not the standard form for a linear equation. So, to fix this, we transfer the 5 to the left side of the equation. And that's all you do, rewrite the 5 on the other side, with one catch. You have to change the symbol. So, if it's a negative number, when you write it on the other side, it will become a positive number. If it was already positive, it will become negative once on the other side.
10x+10-5=0\\ 10x+5=0
Now, once we have the right form, we can calculate the unknown.
Don't forget, when it comes to math, practice makes perfect. Your understanding of this subject will greatly improve if you solve some problems by yourself.
Linear equations with two variables
As we mentioned before, they can also come with multiple variables. For the two-variable ones, the standard form is :
{\color{Green}a} \cdot {\color{Red}x} + {\color{Green}b} \cdot {\color{Red}y} + {\color{Blue}c} = 0
Solving these equations is different than solving the one-variable linear equations. Let me show you an example. Let's say we have the following equation:
-5x + 4y = 12
In this case, the constant is0, so we can just leave it out.
As we mentioned, solving any equation means finding the values for the unknowns that make the equality of the two expressions true. When you have two variables, there is more than one solution.
Now, I am going to give you some potential solutions for this problem, and you can try to find the correct one(s).
(4,5) \\ (-4, -2) \\ (4,13) \\ (5,4)
It is important to remember that these always come in pairs, because you need values of both variables to solve the equation. The first number is alwaysx and the second is alwaysy. So, to find the correct solution(s), we have to test these out to see if they fit the criteria. Let's check them all in order, and see which ones fit.
1. -5 \cdot 4 + 4 \cdot 5 = -20 + 20 = 0 ; \boxed {0 \not = 12} ; \text {(4,5) \textcolor{red}{is not} a solution} \\ 2. -5 \cdot -4 + 4 \cdot -2 = 20 + (-8) = 12 ; \boxed {12 = 12} ; \text {(-4,-2) \textcolor{lime}{is} a solution} \\ 3. -5 \cdot 4 + 4 \cdot 13 = -20 + 32 = 12 ; \boxed {12 = 12}; \text {(4,13) \textcolor{lime}{is} a solution} \\ 4. -5 \cdot 5 + 4 \cdot 4 = -25 + 16 = -9 ; \boxed {-9 \not = 12} ; \text {(5,4) \textcolor{red}{is not} a solution} \\
We have found two solutions for the given equation, (-4,-2) and (4,13). Once again, if we use these two values as the x and y coordinates in a graph, we will get two dots that, when connected, produce a straight line.
Examples
As we saw in the previous examples, they can gain different forms in certain situations. Essentially, if any value is 0, that part can be left out of the equation. Let's see some examples.
0x + 5y + 2 = 0
Since the linear constant for x is 0, that entire part can be left out. This is because when you multiply a number with 0, the product will always be 0, meaning the value of x does not matter. So, we can rewrite this equation as:
0 + 5y + 2 = 0 \\ 5y + 2 = 0
As we can see, this is basically aone-variable linear equation, and it can be treated as such.
I mentioned the two basic forms, but there are many more . For example, the slope-intercept form:
y = mx + y_0
This one is useful for any linear equation where the line produced is not vertical.m is the slope of the line, andy0is the point where the line intercepts with the y axis.
Linear function
Linear equations are connected to linear functions. They can come in many forms, but the most common one is the slope-intercept form. As we mentioned, the general formula for it is
f (x) ={\color{Teal}y}= a \cdot {\color{Green}x} + b
Let me give you an example of this form:
{\color{Teal}y} = 2 \cdot {\color{Green}x} + 2
Now if we start exchanging x for actual numbers we start to see the reason for the name:
\boxed {{\color{Teal}2} = 2×{\color{Green}0} + 2} \\ \boxed {{\color{Teal}4} = 2×{\color{Green}1} + 2} \\ \boxed {{\color{Teal}6} = 2×{\color{Green}2} + 2} \\ \boxed {{\color{Teal}8} = 2×{\color{Green}3} + 2} \\ \boxed {{\color{Teal}10} = 2×{\color{Green}4} + 2}
We can see that the progression of x is 0, 1, 2, 3, 4, and the progression is y is 2, 4, 6, 8, 10. Every time x changes by 1, y changes by 2. This type of progression is linear, hence the name of the equation. This is why the variable x can't have an exponent, and it can't be a root of a number, as the progression would no longer be linear.
Furthermore, if you put this into a graph you will see a vertical line. The vertical line will go through every x and y coordinate (x:1, y:4; x:2, y:6, etc.).
What are systems of linear equations?
A system of linear equations is a set of linear equations that all contain the same variables. A system can have between two and four equations. In order to find the solution for a system, you need to find the values of the unknowns that satisfy every equation in the system. Let me give you an example:
20+10x=0 \\ 8+4x=0
This is a system of two linear equations. In order to solve it, we need to find the value of x that makes the equality of both equations true. For this one, it is pretty simple, considering we only have one variable.
20+10x=0 \\ x = -\frac {20}{10} \\ \boxed {x = -2} \\ 20+10 \cdot (-2)=0 \\ 8+4 \cdot (-2) = 0
This works because this system is consistent, meaning there is at least one solution to it. If a system has no solutions, it is dubbed inconsistent.
These might seem complicated at first, however, the more you practice, the better you will get.
FAQ
How to write a linear equation
They come in many forms, but you will most often run into the one-variable (ax+b=0) and two-variable, (ax+by+c=0) linear equations.
How to graph a linear equation
With our calculator, you can easily graph all your linear equations.
Find All Solutions to the System of Linear Equations Calculator
Source: https://calconcalculator.com/math/linear-equation-calculator/