00:02 Hi, everyone.

00:15 My name is Baruwa R.O. Mrs.

00:18 I'm your master teacher for today.

00:20 And the topic we'll be looking at

00:21 is some autonomous equation,

00:24 one linear and one quadratic.

00:28 At the end of this section,

00:30 you should be able to recall

00:31 linear and quadratic equations.

00:34 Also, you should be able to solve

00:36 some autonomous equation,

00:38 one linear and one quadratic

00:40 using substitution method.

00:42 Firstly, what is linear equation?

00:57 Linear equation is an algebraic equation

01:01 that has between them two variables.

01:05 And it has one exponent.

01:07 It has the power of one.

01:09 And when you plot it on the graph,

01:11 it gives you a straight line.

01:13 Example of linear equation R,

01:16 Y is equals to Mx plus C.

01:19 The power of M here is one.

01:23 So it has one power.

01:25 And the power of S is also one.

01:27 Another example is Causty is equals to 6x plus four.

01:32 The power of X here is one.

01:38 So this one is the example of linear equation.

01:42 Also, what is quadratic equation?

01:44 Quadratic equation is an algebraic equation

01:47 that has a general formula AX square plus BX plus C

01:55 is equals to zero.

01:56 We are A is not equals to zero.

01:59 And the exponent of quadratic equation is two.

02:03 It has two ascetics power.

02:06 Now, solve the quadratic equation

02:09 and linear equation using substitution method.

02:13 Let's look at the following examples.

02:15 Now, looking at this following example,

02:19 example one, two X plus Y is equals to four.

02:26 X square plus XY is equals to minus 12.

02:32 Aceties.

02:34 Two X plus Y is equals to four is equation one.

02:41 Y square plus S is equals to minus 12 is equation two.

02:46 Now, from equation one,

02:52 B, the linear equation, make Y the subject of the formula.

02:55 There, you have Y is equals to four minus two X,

03:00 which is equation three.

03:03 Now, that Y is four minus two X.

03:06 Substitutes Y into equation two.

03:19 And this will now give you X square plus X

03:28 into four minus two X is equals to minus 12.

03:35 Open the bracket.

03:37 You have S square plus four X minus two X square

03:44 is equals to minus 12.

03:47 Aceties.

03:48 Collect like tapes by hiding square minus two X square.

03:56 You have your minus X square plus four X is equals to minus 12.

04:04 You have the X square to be negative.

04:07 So multiply through by minus one.

04:11 So that your X square will be positive.

04:13 And you will form your quadratic equation and solve accordingly.

04:17 You have minus minus one times S square,

04:21 we give you S square minus four X is equals to 12.

04:27 You've already multiplied by minus one.

04:34 Now, equate everything to zero so that you can factorize.

04:45 You have X square minus four X minus 12 is equals to zero.

04:54 Don't forget, anytime you have 12 or an integer

04:59 at the back of equal to sign, when you carry it over,

05:03 it returns to negative, if it is positive.

05:06 But if it is negative, it returns to positive.

05:10 Now, factorize.

05:12 The coefficient of S square is one, and your constant is minus 12.

05:16 So you are looking for a number that you multiply it together.

05:21 It will give you negative 12.

05:23 But when you hide it, it will give you minus four.

05:27 And what is the number?

05:30 The product will be minus six plus two.

05:43 Minus six multiplied by plus two, we give you minus 12.

05:48 And minus six plus two, we give you minus four.

05:54 Now, substitute this factor, so where you have minus four.

06:06 And this one will not become X square minus six X plus two X

06:14 minus 12 is equals to zero.

06:17 Don't forget your grouping, you grouped this together,

06:22 then you are good to go.

06:26 X is common to X square and six X.

06:28 So you have X into bracket X minus six.

06:33 Two is common to two X minus 12.

06:36 And this one will give you plus two into X minus six, likewise.

06:44 X minus six and X minus six is common.

06:47 So you bring out one out of the two.

06:49 There you have X plus two is equals to zero.

06:53 Or X minus six is equals to zero.

06:57 From here, we've got it two factors of S.

07:01 We've got the value of X to be X is equals to minus two.

07:06 And X is equals to six.

07:11 From here now, substitute the value of this X you've

07:16 got into equation three.

07:20 If you can remember, our equation three

07:25 is Y equals to four minus two X.

07:31 So substitute its back.

07:32 Substitute X is equals to minus two into equation three,

07:47 which is Y is equals to four minus two X.

08:04 Your X is minus two.

08:05 There you have Y is equals to four minus two,

08:08 multiplied by minus two.

08:10 And Y is equals to four plus four.

08:14 Don't forget.

08:15 Negative two multiplied by negative two

08:17 will give you plus four.

08:19 So Y is equals to eight.

08:23 After solving, we can see that when X is equals to minus two,

08:33 Y is equals to eight.

08:35 So the coordinate is minus two comma eight.

08:39 X comma Y minus two comma eight.

08:42 All right, now when our X is equals to six,

08:48 we have to solve for Y.

08:52 When X is equals to six, go back to equation three.

08:59 Likewise, you have your Y is equals to four minus six.

09:10 When X is equals to six, there you have Y

09:14 is equals to four minus two into six.

09:20 Y is equals to four minus two as an Y is six.

09:23 So therefore, we now have Y is equals to four minus 12.

09:28 Four minus 12 is equals to minus eight.

09:32 So you have Y is equals to minus eight.

09:37 With these, we have our coordinates to be six comma minus eight.

09:44 That is, when X is equals to six, Y is equals to minus eight.

09:53 That is all about example one.

09:57 Example two now.

10:03 X square plus Y square is equals to 29, equation one.

10:22 And X plus Y is equals to seven, equation two.

10:30 Always step note.

10:32 When you are using substitution method,

10:35 you start from linear equation.

10:37 You concentrate on your linear equation first.

10:40 You make one of the variables subject of the formula.

10:43 If you like, you can make your X the subject of the formula.

10:46 If you like, you can make your Y the subject of the formula.

10:49 So from equation two, now from equation two,

10:54 you make Y the subject of the formula,

10:56 there you have Y is equals to several minus X.

11:00 And this is equation three.

11:03 Substitutes equation three into equation one.

11:07 Substitutes equation three into equation one.

11:12 Then you have X square plus seven minus X square is equals to 29.

11:28 Open the brackets.

11:30 You have X square plus seven minus X.

11:36 Seven minus X is equals to 29.

11:39 And this is something as X square plus seven into seven minus X.

11:50 Minus X into seven minus X is equals to 29.

11:58 X square plus seven multiplied by seven 49 minus several times X,

12:05 seven X minus X times seven minus seven X minus X minus X plus X square is equals to 29.

12:17 Now X square plus X square is equals to two X square minus seven X minus seven X

12:25 to give you minus 14 X. This is 49, which is plus.

12:33 If you break 29 because you want to equate everything to zero,

12:38 plus 29 when you cross equal to side become minus 29 is equals to zero.

12:47 Now from here, 49 minus 29 will give you 20.

12:52 So you have two S square minus 14 X plus 20 is equals to zero.

13:03 Already you have your quadratic equation.

13:07 The coefficient of S square is two and the constant is 20.

13:11 So you think of two numbers.

13:13 Now when you multiply it together, it will give you 40.

13:17 But when you hide it together, it will give you minus 14 X.

13:22 And the as it is, you can divide through by two.

13:28 You know, two, 14, 20 has something in common, which is two.

13:33 So before finding the product of the coefficient of S square and the constant,

13:39 let's divide by two.

13:41 In that case, this now gives X square minus seven X plus 10 is equals to zero.

13:57 Divide through by two.

14:04 Now the coefficient of S square is one, our constant is 10.

14:09 Then we have 10.

14:12 The two numbers that we multiply together that we give you 10 is minus five and minus two.

14:23 And if you hide minus five and minus two together, it will give you minus minus seven.

14:37 Product of minus two and minus five will give you plus 10.

14:42 Some of minus two and minus five will give you minus seven.

14:48 Substitute this into the equation.

14:51 There you have X square minus two X minus five X plus 10 is equals to zero.

15:01 Don't forget your grouping, bring it together.

15:06 What is common X?

15:07 There you factorize X minus two minus five into X minus two is equals to zero.

15:17 Now you have your common factor, which is X minus two and X minus two.

15:21 Bring out one.

15:22 Then you have X minus five is equals to zero.

15:30 X minus two is equals to zero.

15:34 From here, our S is equals to five or S is equals to two.

15:41 Since you've gotten the value of X,

15:44 substitute the value of X into equation three.

15:49 When X is equals to five,

15:54 substitute it into equation three.

16:02 And this we give you.

16:07 Equation three is y equals to seven minus X.

16:16 Y equals to seven minus X.

16:20 So we have y is equals to seven minus five.

16:26 Since our X is five and we have y is equals to two.

16:31 From here, we know that when X is equals to five,

16:34 y is equals to two.

16:37 And the coordinates here is five comma two.

16:42 Also, when X is equals to two, what would be the value of y?

16:51 When X is equals to two,

16:56 y is equals to seven minus two.

16:59 And y is equals to five.

17:01 Therefore, when X is equals to two, y is equals to five.

17:08 And the coordinates will be X comma y,

17:14 which is two comma five.

17:20 Looking at the example three,

17:22 X square minus two, y square is equals to one,

17:28 which is equation one.

17:31 X plus five is equals to--

17:37 excuse me-- X plus y is equals to five equation two.

17:47 Always put into consideration your linear equation.

17:51 So if you make X the subject of the formula here,

17:54 where you have X is equals to five minus y, equation three.

18:01 From here, you solve for X, you substitute the value

18:05 back to equation one, which is a quadratic equation,

18:09 and you solve for y.

18:11 So as it is now,

18:16 substitute equation three into equation one.

18:26 And this will give you five minus y square minus two,

18:37 y square is equals to one.

18:41 Open your brackets.

18:42 You have five minus y.

18:45 Multiply by five minus y minus two, y square is equals to one.

18:53 From here, five into five minus y minus y into five minus y

19:08 minus two, y square is equals to one.

19:13 Five multiply by five, 25 minus five y minus five y

19:19 plus y square minus two, y square is equals to one.

19:26 As it is, this is y square.

19:28 This is minus two, y square.

19:29 We collect it together.

19:30 As if you hide it together, you have minus y square.

19:35 Minus five y minus five y will give you minus 10 y.

19:41 This is 25 plus 25.

19:45 We want to equate everything to zero,

19:47 so that we can have our quadratic equation.

19:49 So bring it back, you have minus one.

19:51 It's equals to zero.

19:53 Minus y square minus 10 y plus 25 minus one,

20:04 we give you plus 24 is equals to zero.

20:08 Note, the coefficient of y square must be positive.

20:13 And here we have negative.

20:14 So you multiply through by minus one.

20:16 And this give you y square plus 10 y minus 24 is equals to zero.

20:23 Multiply true by minus one.

20:33 Now you cannot factorize.

20:37 The coefficient of y is one, you have your constant

20:39 to be minus 24.

20:41 You take off two number that when you multiply it,

20:43 it will give you negative 24.

20:44 But when you hide it together, it will give you 10.

20:47 And the product will be plus 12 minus two.

20:59 If you hide, if you hide, plus 20,

21:07 you have a minus two together.

21:10 It will give you positive 10.

21:14 So with this, substitute plus positive 12

21:17 and negative 2 to equation, then we have y square plus 12y

21:29 minus 2y minus 24 is equals to zero.

21:35 Don't forget your grouping.

21:39 When you group this together, you have y

21:43 into y plus 2, what is common here?

21:47 Minus two into minus two into y plus 12 is equals to zero.

21:58 Y multiplied by y will give you y square.

22:02 Y multiplied by 12 will give you plus 12y.

22:05 Minus two times y will give you minus two y,

22:08 and minus two times plus 12 will give you minus 24.

22:12 We have same factor here.

22:14 So you bring out one, you have y plus 12 is equals to zero,

22:20 or y minus two is equals to zero.

22:23 And this one makes us know that y is equals to minus 12,

22:32 or y is equals to two.

22:35 Since we've got the value of y, we can now solve for s.

22:39 Substitutes y is equals to minus 12 into equation three.

22:45 Recall that your equation three is--

23:03 x is equals to five minus y.

23:05 So since we have our s is equals to five minus y,

23:08 and now that we know the value of y is substitutes.

23:12 Now x is equals to five minus y.

23:16 When y is minus 12, we have x is equals to five

23:23 into minus 12, five minus 12.

23:29 And this one we give us s is equals to five plus 12.

23:32 And x is equals to 17.

23:35 From here, when x is equals to 17,

23:42 y is equals to minus 12.

23:45 And the coordinate is 17, minus 12.

23:50 Also, when x is equals to three, we've

24:01 got to the true value of x to be minus 12.

24:06 We've got to the true value of y to be minus 12 and plus two.

24:09 So when x is equals to plus two, we have x is equals to five

24:21 plus two.

24:30 s is equals to seven.

24:33 Therefore, when x is seven, y is equals to two.

24:39 And the coordinate is seven comma two.

24:44 Looking at example four, x squared minus y squared is equals

24:49 to minus 45, equation one.

24:52 y minus 5s is equals to three, equation two.

24:56 You solve accordingly.

24:57 Then you will get the value of x and y.

25:00 So when y is equals to two, from equation three,

25:04 x is equals to five minus y.

25:08 Substitute the value of y, your f is equals to five minus two.

25:14 And the x, we give us three.

25:17 Therefore, x is equals to three when y is equals to two.

25:24 And we have the coordinate three comma two.

25:28 Looking at the fourth example, x squared minus y squared

25:30 is equals to minus 45.

25:33 And y minus 5, plus is equals to three.

25:36 If you solve it, you will get the value of x

25:41 to be three of a four, or x is equals to minus two.

25:45 Substitute the value of x to your quadratic equation three.

25:50 Then you get your value of y to be 27 over four,

25:54 or 6 over number three over four.

25:58 Also, when you are solving, you can solve for x first.

26:04 Then you substitute.

26:05 And if you like, you can solve for y.

26:07 You are sequence to get the same answer.

26:11 Evaluation, I want you to solve these two questions.

26:16 Solve the following piece of some autonomous equation

26:19 using substitution method only.

26:22 x squared minus y squared is equals to 21.

26:26 x plus y is equals to seven.

26:28 And now x squared minus y squared is equals to 50

26:31 on number three over four.

26:33 Three x plus y is equals to minus 10 on number one over two.

26:39 Thanks for watching.

26:41 I am Barua R.O. Mrs. Till next time.