Two linear equations with two unknowns

In general, a system of linear equations placed contiguously or written one below the other, either within braces or without symbols, is called a system of linear equations.
When we do not know the value of $X$ and $Y$, we can call them unknowns and understand that it is a system of linear equations with two unknowns.
Anyway, when the question says "given a system of equations" you have to know that it refers to a system of equations in which the unknowns must meet all the imposed conditions.

First, let's start by asking what is a linear equation?
A linear equation is an equation of the type:
$y=ax+b$

For example:   $30=7X-19$

The $X$ is our unknown and the linear equation describes a certain condition or property that this $X$ must fulfill.
We are not using the term condition just because, you will understand shortly.
If we solve our equation, transpose members and isolate the $X$ we will find that:
$X=7$
when the $X$ equals $7$ it is indeed fulfilling the property or condition asked by the equation.
When we place it in the equation we obtain equality in both members.
Also in the equation:
$13-4x=2x-5$
The unknown $X$ must fulfill a certain property or condition for there to be equality in the members: Solve the equation.
If we transpose members and isolate the $X$ we will discover that the only $X$ that fulfills the condition and solves the equation is $3$.
So, after we have talked about equations with a single unknown, we will move on to linear equations with two unknowns.

The linear equation with two variables is also in the form: 
$y=ax+b$
Only this time, we do not know either $X$ or $Y$.
In fact, a linear equation with two variables is an equation in which we must find the two specific unknowns that meet the required condition in the equation.
Did that confuse you?
Let's see an example:
In this equation $x+y=5$
The condition or property is that the sum of the two unknowns $X+Y$ equals $5$.
Indeed, to solve the equation we must find a pair of numbers that together equal $5$.
Actually, the solution to this equation can be any pair of numbers whose sum is $5$, and therefore, there is not just one solution, but infinitely many solutions.

The first step is to isolate a variable. Select the variable you prefer and isolate it by leaving it alone on one side of the equation.
In our example, we will isolate $X$ and we will obtain:
$x=5-y$
The second step is to replace the non-isolated variable (in this case $Y$) with any number you want, this way you will discover the $X$ that meets the required condition.
For example,
let's put $y=2$
and we will get $x=3$
Notice, we can put any number we want, therefore, this equation does not have a single solution, but infinite solutions!

Correct solutions for this example could be $y=4 ,x=1$ or $y=-1 ,x=6$ or $y=0.1, x=4.9$ and so on.
Any pair that meets the condition $x=5-y$
will be a correct solution since both variables would make up the equation.

If we look at the values of $X$ and $Y$ as points and mark them on the Cartesian plane, when we draw a line between them we will discover what the equation looks like. Now, after having reviewed the topic of linear equations, we will move on to see what a system of equations is.

Don't stress over the word system, try to think of a system as a collection or group of conditions that must be met simultaneously.
What does it mean?
Observe the system of equations presented below:
$​​​​​​​x+y=5$
$y-x=3$

In this system, there are two equations with two unknowns, $X$ and $Y$.

The first condition for the system to be satisfied is that the sum of the two variables equals $5$, the second condition is that the difference between $Y$ and $X$ is $3$.
To solve this equation, we must find $X$ and $Y$ that meet both conditions simultaneously.
That is, the solution to a system of equations lies in finding the specific unknowns that meet all the conditions of the system.
Note that, $X$ and $Y$ that meet only one condition of the system would not be a correct answer!
For example, if $x=3$ and $y=2$ the first condition is met since $2+3=5$ but, when placing the data in the second equation we get $-2-3=3$ which is clearly wrong.
Therefore, $x=3 , y=2$ is absolutely not the correct answer and, in fact, you wouldn't even receive a point for it.
The only solution for the system of equations in this example is $X=1 ,Y=4.$
How did we arrive at this solution?
Next, we will teach you all the methods you need to know to solve systems of linear equations with two unknowns.
We will start with the graphical method.

You can solve a system of linear equations with two variables through the graphical method, in a way that is very easy to understand.
When we want to solve a system of linear equations using the graphical method, we will refer to the equations as functions and graphically represent the two equations on the Cartesian plane.

To help you understand how a system of equations is solved with the graphical method, we will use the system we previously introduced:
$x+y=5$
$y-x=3$

First step: We will refer to each equation as a function, isolate the $Y$ and make a value table for each function/equation.
Don't be scared, the value table for a function is a simple table of $X$ and $Y$ that we can create very easily.
We will show it as follows: 
$x+y=5$

$y=3+x$

After drawing the value table, we will fill one of the unknowns with values randomly.
We recommend placing in the unknown $X$ the values $0$, $1$, and $2$. You should know that it doesn't matter which numbers you choose to put in $X$, we recommend $0$, $1$, and $2$ because this is comfortable as a working system.
After noting in the $X$ column the chosen values, we will add one by one the values of the function, we will discover the $Y$ and fill the $Y$ column of the table accordingly. In this same way, we will do it with the two functions separately.

For example:
in the function    
$y=5-x$
let's see what Y we will get when we place $X=0$
$y=5-0$
$y=5$
We will continue checking what happens when $X=1$
$y=5-1$
$y=4$
And so on.
We will fill the tables respectively:
$y=5-x$

$y=3+x$

The second step:
After each equation has been given a table of values that meets the conditions of its equation, we will draw a Cartesian plane.
The horizontal axis or x-axis, $X$
The vertical axis or y-axis, $Y$
We will choose an equation/function to start with and represent it on the Cartesian plane according to the table we have created.
After marking all the points of the function, we will connect them to form a line.
Note that each function has a different table, therefore, to avoid confusion, it is necessary to draw the line between the points we have already marked before we continue representing the new points of the second function on the plane.
After drawing the line between the three points, we can continue marking the following ones.
We will also connect these points to form a line.

Remember that in the first step you had isolated $Y$ to be able to work comfortably. To avoid confusion, write next to each original equation on the Cartesian plane the isolated equation. This way, you will be able to know which isolated equation is equal to the original.
It is always advisable to use different colors for each function on the Cartesian plane. Paint the equation and its graph in the same color. This way, you can easily distinguish the functions and see them in their corresponding color on the graph.
Also, note next to each line created the matching equation to easily know which line belongs to each function.

System of two linear equations with two variables

Perfect! Now, after having represented the equations through the graphical method, you can easily find the solution to the equation.
Do you remember that we had said that the solution to a system of equations is, in fact, a pair of $X$ and $Y$ that simultaneously meet the conditions of the equations?
Could you now evaluate which point on the graph meets both conditions?
Clearly, it is the point of intersection of the graphs! The point of intersection coincides with both equations which implies that the $X$ and the $Y$ meet both required conditions.
In the illustration, you can see the values of $X$ and $Y$ at the point of intersection and thus find the solution through the graphical method.
In our example, the solution to the system of equations is $X=1$ $Y=4$
which is, clearly, the point of intersection of the functions we plotted.

System of two linear equations with two variables

Useful Information:

In two linear equations with two unknowns, it can happen that they have infinite solutions when the lines overlap each other or that they have no solution if the lines were parallel and, therefore, would never intersect, or that they have a single solution if the lines intersect at only one point.
Linear equations with two unknowns cannot intersect at more than one point, therefore, there cannot be $2$ or $3$ solutions.
(except in the case where the lines overlap coinciding at all their points and then there would be infinite solutions)

Now that you know what a linear equation with two unknowns is and you know what a system of linear equations is, and besides, you have learned the graphical method to find the solution, all that remains for you to do is practice the method so you can carry it out quickly and naturally.
Obviously, there are other methods to find the solution of the system of linear equations with two unknowns, we are here to present them to you.
In certain cases, you will be able to decide which method to use and clearly, you will always arrive at the same result, in others, you might be asked in an exam to do it with a specific method.
Either way, with Tutorela you will have all the necessary tools to reach the appropriate solution.

If you are interested in this article, you might also be interested in the following articles

• Algebraic solution for a system of linear equations with two unknowns
• Unknowns and algebraic expressions
• First-degree equations with one unknown
• What is the unknown of a mathematical equation?
• Linear equation with two unknowns
• Solution with graphical method for a system of linear equations with two unknowns
• Solution with the substitution method for systems of two linear equations with two unknowns
• Solution with the equalization method for systems of two linear equations with two unknowns
• Solving verbal problems with a system of linear equations

In the blog of Tutorela you will find a wide variety of mathematics articles