The Art of Solving Equations: A Step-by-Step Guide

In mathematics, solving equations is an essential skill that involves finding the values of variables that make the equation true. The process of solving equations can seem daunting at first, but with a clear understanding of the concepts and techniques involved, anyone can master it. In this article, we will explore the step-by-step process of solving linear equations, specifically those in the form of X plus 2y equals 6.

First, let us take a look at one of these equations, where we set X to be equal to zero as an example. The equation becomes zero plus two y is equal to six. To solve for Y, we need to isolate the variable Y on one side of the equation. We can do this by dividing both sides of the equation by 2. This gives us Y equals six divided by 2, which simplifies to Y equals three. Therefore, when X is set to zero, the solution to the equation is X comma 3.

Next, let's consider another example where we set Y to be equal to zero. The equation becomes X plus two times zero is equal to six. Simplifying this equation, we get X equals six. Therefore, when Y is set to zero, the solution to the equation is 6 comma 0.

Now, let us take it one step further and find a third solution to the equation by setting Y to be equal to one. The equation becomes X plus two times one is equal to six. Simplifying this equation, we get X equals four. Therefore, when Y is set to one, the solution to the equation is 4 comma 1.

Finally, let us consider an example where X is set to two. The equation becomes two plus two Y is equal to six. To solve for Y, we need to isolate the variable Y on one side of the equation. We can do this by subtracting two from both sides of the equation and then dividing both sides by 2. This gives us Y equals four divided by two, which simplifies to Y equals two. Therefore, when X is set to two, the solution to the equation is 2 comma 2.

The key to solving equations is to be able to isolate the variables on one side of the equation and simplify the expressions. By using basic algebraic operations such as addition, subtraction, multiplication, and division, we can find the values of X and Y that make the equation true.

In conclusion, solving linear equations in the form of X plus 2y equals 6 involves isolating the variables on one side of the equation and simplifying the expressions. By applying basic algebraic operations and following a step-by-step approach, anyone can master the art of solving equations.

"WEBVTTKind: captionsLanguage: engood morning students today we are going to start our next chapter chapter number four that is linear equations in two variables so in the previous class you had learned about linear equations in one variable so what are the linear equation equation having degree 2 degree 1 is known as a linear equation for example 4x plus 3 is equal to 0 so this equation is a linear equation because here degree is 1 and you know very well to our data how can we find that degree so as you see that here power of X is 1 so highest power of X is 1 so our degree is also 1 so degree equation having degree 1 is known as a linear equation clear so now we learnt about the linear equations in two variables so as you see for y plus 1 this is also a linear equation but in one variable here variable is y also you know very well that in linear equation each equation have a unique solution means one and only one solution so now in this chapter we will discuss that this a linear equation in two variables have a solution if yes then is it unique so what does the solution look like on the Cartesian plane okay so we will discuss these properties in the in this chapter ok so first of all and we will discuss that linear equation in two variables for example 4x plus 3y is equal to 0 so this is a linear equation but in two variables we use here two variables x and y so and also you see that a degree is 1 so this is linear equation in two variables so an equation which can be put in the form ax plus B Y plus C is equal to zero where a B and C these are real numbers real numbers and a and B are not both 0 dohno 0 neonate a ticket this is called the linear equation in two variables okay clear so linear equation this is the standard form of linear equation in two variables you have to remember this standard form okay now let us discuss the example the first example is write each of the following equations in the form of ax plus B y plus C is equal to 0 and indicate the values of a B and C in each case so we have to write this equation in a X plus B y plus c f-- in this form okay and also we have to indicate the values of a B and C so first of all let us take the example number one first part 2x plus 3y is equal to four point three seven so first of all we have to write this in this form so ax 2 X plus B y 3y now take this on the left hand side it becomes negative four point three seven is equal to zero okay yukio my school left hand side pillai i two aside a multipass cabbage' zero okay so as you see this is in the in this form ax plus B by plus C or you can write it as 2x plus 3y + bracket minus four point three seven okay so now we have to find the value of a B and C so compare this equation with this with the standard equation so as you know the standard equation is ax plus B y plus C is equal to zero so compare this as you know this is a this is B and this is C so a is 2 B is 3 and C is minus four point three seven okay in this way we can compare the equation let us take another example that is 2x is equal to Y okay first of all we take Y on left hand side 2x it becomes negative because here it is of positive so 2x minus y is equal to zero or also we can write it as 2x plus minus y is equal to zero looks at MSM also we can write it as 2x plus minus y plus 0 is equal to 0 as Aleksic Nana 0 add here subtract the colloquy factor Tony padega now we compare it with the standard equation above name is go ax plus b bi plus DG formally clea now we write the value of a b c by comparing it with the standard equation b y plus c okay so as you see value of phase 2 value of B is minus 1 and value of C is zero a is 2 B is minus 1 is 0-1 guess I have because we can write a minus y as minus 1 into whiner okay so that's why value of B is minus 1 play the next example is right each of the following as an equation in two variables now we have to write it in as equation in two variables let us take the example example number two first part X is equal to minus five so we have to represent it into two variables okay so it can be written as ax we can write it as 1 into X 1 into X licks at the Amoco ticket looking 1 into X answer axial okay so also we can write it as 1 into X plus 0 into y square simply accept a UK 0 into y answer Galaga 0 ok so also now take five on this side it become positive plus 5 is equal to 0 so in this way we can represent the given equation into the two variables so here two variables one is X and second is why ok let us take another example 2 X is equal to 3 ok so we can write it as 2 X plus 0 into y because 0 into y answer is 0 is equal to 3 now take 3 on this side 2 X plus 0 into y minus 3 is equal to 0 so in this way we can represent ok now here is exercise 4.1 the first question is the cost notebook is twice the cost of pan chicken write a linear equation in two variables to represent this statement we have to represent this statement by writing it in our linear equation okay so how we write it let us write first question exercise 4.1 first of all this is that coast of notebook is twice the cost of pan heat your notebook Caicos there whoop and Caicos toka doubler okay so first of all let us take let Coast off notebook B X rupees X okay and coast of pan is equal to rupees why okay now according to the statement just suggest him come but the JA reimbursable silicone a ticket according to statement cost of normal host of not book is what is the meaning office his means equal device device means two times the cost of pen host off pen so what is the cost of notebook cost of notebook is X is equal to two x what is the cost of pen cost of pennies Y so X is equal to 2y now take this on this side x minus 2 y is equal to 0 in this way we can write it okay and in the next example express the following linear equations in the form ax plus B y plus C equal to 0 and indicate the values of a B and C so this exam this question is same as example number one just an example first Katy Wesley opening a question gonna be cool same clear now we will discuss about solution of a linear equation okay so you have seen that every linear equation in one variable has a unique solution everyone knows about it now so now what can you say about solution of linear equation in two variables what we can say about it so as you know that the linear equation in two variables there are two variables in the equation for example 4x plus 3y is equal to 0 so you see that there are two variables x and y ok so solution means a pair of values because we have two variables so its solution means pair of values 1 for X and a second one for y okay which satisfy the given equation so here X is equal to 3 and Y is equal to minus 4 is a solution why because when we substitute here value of x and y in this equation its answer is 0 for example when we substitute the value 4 into 3 plus 3y is minus 4 so three fours are 12 minus 3 plus minus minus 3 fours are 12 is equal to 0 okay so that's why X is equal to 3 and Y is equal to minus 4 is a solution of this equation so as you see that X is equal to minus 3 and the Y is equal to 4 this is also a solution of this equation ok so you have seen at least two solution of this equation one is 3 comma minus 4 and another is minus 3 comma 4 so there is no end to the different solutions of a linear equation in two variables so that is a linear equation in two variables has infinitely many solutions means we can find many solution of a given equation ok you can also find more than two solution ok so therefore we can find infinitely many solutions ok so now this now we will discuss the example example number three find four different solutions of the equation X plus 2y is equal to 6 now we have to find the four different solution of this equation now how can we find the solution of this equation so let us solve X plus 2y is equal to 6 first of all take of the value of one variable as 0 okay you may take a 0 1 2 3 4 which fill a sector ticket but a variable Killaloe egg variable cap method of say assume Collini and so on CMR per second variable key value our Jacky ticket or assume cancer care sector Kobe value assume curse actor just as 0 1 2 3 4 kuch bhi ticket so for example here I take let X is equal to 0 sports car thank you agar XQ value 0 Thea chicken so put the value here zero plus 2y is equal to 6 2 y is equal to 6 y is equal to 6 divided by 2 which is equal to 3 so when we put value of x of y is 3 chicken am I am people to Kirk a the co six are they Co x is 0 plus 2 into y is 3 0 plus 2 x 3 6 6 plus 0 6 2 bundy Anna it means 0 comma 3 satisfied Cara this is equation Co okay so next above y b0 lace up their ticket let y is equal to 0 so now put this value here x plus 2 x 0 is equal to 6 so x + 2 x 0 answer is 0 is equal to 6x is 6 so when y is 0 X is 6 second is this one 6 comma 0 first resolution is 0 comma 3 second solution is 6 comma 0 okay now we can find the third solution also by taking you may take value of x or y you see can be less active chicken let us take now value of y is 1 okay so put this value here x + 2 x 1 is equal to 6 - x one answer is 2 is equal to 6 X is equal to 6 minus 2 which is equal to 4 so third solution is 4 comma 1 clear in this way we can find the fourth solution also here I will solve it so as you see now take any value of y for example or you can also take the value of x let us take value of x is 2 okay so put here 2 plus 2 y is equal to 6 now take this to on right-hand side no y is equal to 6 minus 2 2 y is equal to 4 y is equal to 4 upon 2 which is equal to 2 so the fourth solution is , - so we get the four solutions that is zero three six zero four one and two Puma - okay clear to everyone how to find the solutions here we have the exercise so in the first question we have to find the solutions are unique to solution or infinite many solution ticket so we can find it very easily okay so next is right for solution of for each of the following equations so in this question we have to just write the four solutions yes MA Abby example discussed here so similarly equation vesical not clear so the third question is check which of the following are the solution of equation this and which are known the equation may have nebith on a key cone saw in my sequence a solution is equation K hair or consignee air ticket to a APNIC SS alternate for example 0 comma 2 where my bus ticket to X value of x is 0 and value of y is 2 so who is simply we have to put this in this equation okay if we get answer 4 then we say that it is a solution if answer is not 4 then we can say that it is not our solution so let us discuss the solution is given 0 comma 2 okay first of all we take left hand side that is X minus 2 y value of x is 0 value of y is totes put here 0 minus 2 into 2 0 minus two twos are four which is equal to minus 4 so as you see which is not equal to 4 which is not equal to right hand side so we say that 0 comma 2 is not a solution of X minus 2 y is equal to 4 this equation clear to everyone okay in the next question find the value of K if X is equal to 2 and Y is equal to 1 is a solution of the equation this one ok now it is given that X is equal to 2 and Y is equal to 1 is the solution of this equation and we have to find the value of K so we simply put the value of x and y here okay so and to find the value of K so you have to try these questions cleargood morning students today we are going to start our next chapter chapter number four that is linear equations in two variables so in the previous class you had learned about linear equations in one variable so what are the linear equation equation having degree 2 degree 1 is known as a linear equation for example 4x plus 3 is equal to 0 so this equation is a linear equation because here degree is 1 and you know very well to our data how can we find that degree so as you see that here power of X is 1 so highest power of X is 1 so our degree is also 1 so degree equation having degree 1 is known as a linear equation clear so now we learnt about the linear equations in two variables so as you see for y plus 1 this is also a linear equation but in one variable here variable is y also you know very well that in linear equation each equation have a unique solution means one and only one solution so now in this chapter we will discuss that this a linear equation in two variables have a solution if yes then is it unique so what does the solution look like on the Cartesian plane okay so we will discuss these properties in the in this chapter ok so first of all and we will discuss that linear equation in two variables for example 4x plus 3y is equal to 0 so this is a linear equation but in two variables we use here two variables x and y so and also you see that a degree is 1 so this is linear equation in two variables so an equation which can be put in the form ax plus B Y plus C is equal to zero where a B and C these are real numbers real numbers and a and B are not both 0 dohno 0 neonate a ticket this is called the linear equation in two variables okay clear so linear equation this is the standard form of linear equation in two variables you have to remember this standard form okay now let us discuss the example the first example is write each of the following equations in the form of ax plus B y plus C is equal to 0 and indicate the values of a B and C in each case so we have to write this equation in a X plus B y plus c f-- in this form okay and also we have to indicate the values of a B and C so first of all let us take the example number one first part 2x plus 3y is equal to four point three seven so first of all we have to write this in this form so ax 2 X plus B y 3y now take this on the left hand side it becomes negative four point three seven is equal to zero okay yukio my school left hand side pillai i two aside a multipass cabbage' zero okay so as you see this is in the in this form ax plus B by plus C or you can write it as 2x plus 3y + bracket minus four point three seven okay so now we have to find the value of a B and C so compare this equation with this with the standard equation so as you know the standard equation is ax plus B y plus C is equal to zero so compare this as you know this is a this is B and this is C so a is 2 B is 3 and C is minus four point three seven okay in this way we can compare the equation let us take another example that is 2x is equal to Y okay first of all we take Y on left hand side 2x it becomes negative because here it is of positive so 2x minus y is equal to zero or also we can write it as 2x plus minus y is equal to zero looks at MSM also we can write it as 2x plus minus y plus 0 is equal to 0 as Aleksic Nana 0 add here subtract the colloquy factor Tony padega now we compare it with the standard equation above name is go ax plus b bi plus DG formally clea now we write the value of a b c by comparing it with the standard equation b y plus c okay so as you see value of phase 2 value of B is minus 1 and value of C is zero a is 2 B is minus 1 is 0-1 guess I have because we can write a minus y as minus 1 into whiner okay so that's why value of B is minus 1 play the next example is right each of the following as an equation in two variables now we have to write it in as equation in two variables let us take the example example number two first part X is equal to minus five so we have to represent it into two variables okay so it can be written as ax we can write it as 1 into X 1 into X licks at the Amoco ticket looking 1 into X answer axial okay so also we can write it as 1 into X plus 0 into y square simply accept a UK 0 into y answer Galaga 0 ok so also now take five on this side it become positive plus 5 is equal to 0 so in this way we can represent the given equation into the two variables so here two variables one is X and second is why ok let us take another example 2 X is equal to 3 ok so we can write it as 2 X plus 0 into y because 0 into y answer is 0 is equal to 3 now take 3 on this side 2 X plus 0 into y minus 3 is equal to 0 so in this way we can represent ok now here is exercise 4.1 the first question is the cost notebook is twice the cost of pan chicken write a linear equation in two variables to represent this statement we have to represent this statement by writing it in our linear equation okay so how we write it let us write first question exercise 4.1 first of all this is that coast of notebook is twice the cost of pan heat your notebook Caicos there whoop and Caicos toka doubler okay so first of all let us take let Coast off notebook B X rupees X okay and coast of pan is equal to rupees why okay now according to the statement just suggest him come but the JA reimbursable silicone a ticket according to statement cost of normal host of not book is what is the meaning office his means equal device device means two times the cost of pen host off pen so what is the cost of notebook cost of notebook is X is equal to two x what is the cost of pen cost of pennies Y so X is equal to 2y now take this on this side x minus 2 y is equal to 0 in this way we can write it okay and in the next example express the following linear equations in the form ax plus B y plus C equal to 0 and indicate the values of a B and C so this exam this question is same as example number one just an example first Katy Wesley opening a question gonna be cool same clear now we will discuss about solution of a linear equation okay so you have seen that every linear equation in one variable has a unique solution everyone knows about it now so now what can you say about solution of linear equation in two variables what we can say about it so as you know that the linear equation in two variables there are two variables in the equation for example 4x plus 3y is equal to 0 so you see that there are two variables x and y ok so solution means a pair of values because we have two variables so its solution means pair of values 1 for X and a second one for y okay which satisfy the given equation so here X is equal to 3 and Y is equal to minus 4 is a solution why because when we substitute here value of x and y in this equation its answer is 0 for example when we substitute the value 4 into 3 plus 3y is minus 4 so three fours are 12 minus 3 plus minus minus 3 fours are 12 is equal to 0 okay so that's why X is equal to 3 and Y is equal to minus 4 is a solution of this equation so as you see that X is equal to minus 3 and the Y is equal to 4 this is also a solution of this equation ok so you have seen at least two solution of this equation one is 3 comma minus 4 and another is minus 3 comma 4 so there is no end to the different solutions of a linear equation in two variables so that is a linear equation in two variables has infinitely many solutions means we can find many solution of a given equation ok you can also find more than two solution ok so therefore we can find infinitely many solutions ok so now this now we will discuss the example example number three find four different solutions of the equation X plus 2y is equal to 6 now we have to find the four different solution of this equation now how can we find the solution of this equation so let us solve X plus 2y is equal to 6 first of all take of the value of one variable as 0 okay you may take a 0 1 2 3 4 which fill a sector ticket but a variable Killaloe egg variable cap method of say assume Collini and so on CMR per second variable key value our Jacky ticket or assume cancer care sector Kobe value assume curse actor just as 0 1 2 3 4 kuch bhi ticket so for example here I take let X is equal to 0 sports car thank you agar XQ value 0 Thea chicken so put the value here zero plus 2y is equal to 6 2 y is equal to 6 y is equal to 6 divided by 2 which is equal to 3 so when we put value of x of y is 3 chicken am I am people to Kirk a the co six are they Co x is 0 plus 2 into y is 3 0 plus 2 x 3 6 6 plus 0 6 2 bundy Anna it means 0 comma 3 satisfied Cara this is equation Co okay so next above y b0 lace up their ticket let y is equal to 0 so now put this value here x plus 2 x 0 is equal to 6 so x + 2 x 0 answer is 0 is equal to 6x is 6 so when y is 0 X is 6 second is this one 6 comma 0 first resolution is 0 comma 3 second solution is 6 comma 0 okay now we can find the third solution also by taking you may take value of x or y you see can be less active chicken let us take now value of y is 1 okay so put this value here x + 2 x 1 is equal to 6 - x one answer is 2 is equal to 6 X is equal to 6 minus 2 which is equal to 4 so third solution is 4 comma 1 clear in this way we can find the fourth solution also here I will solve it so as you see now take any value of y for example or you can also take the value of x let us take value of x is 2 okay so put here 2 plus 2 y is equal to 6 now take this to on right-hand side no y is equal to 6 minus 2 2 y is equal to 4 y is equal to 4 upon 2 which is equal to 2 so the fourth solution is , - so we get the four solutions that is zero three six zero four one and two Puma - okay clear to everyone how to find the solutions here we have the exercise so in the first question we have to find the solutions are unique to solution or infinite many solution ticket so we can find it very easily okay so next is right for solution of for each of the following equations so in this question we have to just write the four solutions yes MA Abby example discussed here so similarly equation vesical not clear so the third question is check which of the following are the solution of equation this and which are known the equation may have nebith on a key cone saw in my sequence a solution is equation K hair or consignee air ticket to a APNIC SS alternate for example 0 comma 2 where my bus ticket to X value of x is 0 and value of y is 2 so who is simply we have to put this in this equation okay if we get answer 4 then we say that it is a solution if answer is not 4 then we can say that it is not our solution so let us discuss the solution is given 0 comma 2 okay first of all we take left hand side that is X minus 2 y value of x is 0 value of y is totes put here 0 minus 2 into 2 0 minus two twos are four which is equal to minus 4 so as you see which is not equal to 4 which is not equal to right hand side so we say that 0 comma 2 is not a solution of X minus 2 y is equal to 4 this equation clear to everyone okay in the next question find the value of K if X is equal to 2 and Y is equal to 1 is a solution of the equation this one ok now it is given that X is equal to 2 and Y is equal to 1 is the solution of this equation and we have to find the value of K so we simply put the value of x and y here okay so and to find the value of K so you have to try these questions clear\n"