Page 1 PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now, I shall be two times as old as you will be”. Represent this situation algebraically and graphically. Solution: Step1: Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. Step2: Five years ago Maria’s age was(x – 5) years Her Son’s age was (y-5) years Step3: Two years later Maria’s age will be(x + 2) years Her son’s age will be (y + 2) years Step4: Given, 5 years ago Maria’s age = 2 times her son’s age then ? ) 5 ( 2 5 ? ? ? y x ? 0 5 2 ? ? ? y x Step5: Given, 2 years hence, Maria’s age = 2 times her son’s age then, ? ) 2 ( 2 2 ? ? ? y x ? 0 2 2 ? ? ? y x Thus, algebraic representation of the given situations is 0 5 2 ? ? ? y x ………. (I) 0 2 2 ? ? ? y x ………… (II) Step6: To obtain equivalent graphical representation we find two points on the line representing each equation. That is we find two solutions of each equation. Now, 2 2 ? ? y x ……………… FROM (II) ? 2 2 ? ? x y Therefore, 0 ? x ? 1 2 2 0 ? ? ? ? y 2 ? x ? 0 2 2 2 ? ? ? y Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Page 2 PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now, I shall be two times as old as you will be”. Represent this situation algebraically and graphically. Solution: Step1: Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. Step2: Five years ago Maria’s age was(x – 5) years Her Son’s age was (y-5) years Step3: Two years later Maria’s age will be(x + 2) years Her son’s age will be (y + 2) years Step4: Given, 5 years ago Maria’s age = 2 times her son’s age then ? ) 5 ( 2 5 ? ? ? y x ? 0 5 2 ? ? ? y x Step5: Given, 2 years hence, Maria’s age = 2 times her son’s age then, ? ) 2 ( 2 2 ? ? ? y x ? 0 2 2 ? ? ? y x Thus, algebraic representation of the given situations is 0 5 2 ? ? ? y x ………. (I) 0 2 2 ? ? ? y x ………… (II) Step6: To obtain equivalent graphical representation we find two points on the line representing each equation. That is we find two solutions of each equation. Now, 2 2 ? ? y x ……………… FROM (II) ? 2 2 ? ? x y Therefore, 0 ? x ? 1 2 2 0 ? ? ? ? y 2 ? x ? 0 2 2 2 ? ? ? y Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step7: Plot the points A (0,-1) and B (2, 0) on a graph paper. Draw a line passing through the points A and B. Then, the line AB represents the equation 0 2 2 ? ? ? y x Step8: Also, consider 0 5 2 ? ? ? y x ? 2 5 ? ? x y Therefore, 1 ? x ? 3 2 5 1 ? ? ? y 3 ? x ? 4 2 5 3 ? ? ? y Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: x 1 3 y 3 4 Step9: Plot the points P(1,3) and Q(3,4) on the same graph paper. Draw a line passing through the point P and Q. Then the PQ represents the equation 0 5 2 ? ? ? y x We observe that line do not intersect anywhere. x 0 2 y -1 0 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Page 3 PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now, I shall be two times as old as you will be”. Represent this situation algebraically and graphically. Solution: Step1: Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. Step2: Five years ago Maria’s age was(x – 5) years Her Son’s age was (y-5) years Step3: Two years later Maria’s age will be(x + 2) years Her son’s age will be (y + 2) years Step4: Given, 5 years ago Maria’s age = 2 times her son’s age then ? ) 5 ( 2 5 ? ? ? y x ? 0 5 2 ? ? ? y x Step5: Given, 2 years hence, Maria’s age = 2 times her son’s age then, ? ) 2 ( 2 2 ? ? ? y x ? 0 2 2 ? ? ? y x Thus, algebraic representation of the given situations is 0 5 2 ? ? ? y x ………. (I) 0 2 2 ? ? ? y x ………… (II) Step6: To obtain equivalent graphical representation we find two points on the line representing each equation. That is we find two solutions of each equation. Now, 2 2 ? ? y x ……………… FROM (II) ? 2 2 ? ? x y Therefore, 0 ? x ? 1 2 2 0 ? ? ? ? y 2 ? x ? 0 2 2 2 ? ? ? y Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step7: Plot the points A (0,-1) and B (2, 0) on a graph paper. Draw a line passing through the points A and B. Then, the line AB represents the equation 0 2 2 ? ? ? y x Step8: Also, consider 0 5 2 ? ? ? y x ? 2 5 ? ? x y Therefore, 1 ? x ? 3 2 5 1 ? ? ? y 3 ? x ? 4 2 5 3 ? ? ? y Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: x 1 3 y 3 4 Step9: Plot the points P(1,3) and Q(3,4) on the same graph paper. Draw a line passing through the point P and Q. Then the PQ represents the equation 0 5 2 ? ? ? y x We observe that line do not intersect anywhere. x 0 2 y -1 0 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 2. A shopkeeper sold 4 chocolates and 5 lollipops for Rs 8 to a customer. A little lateranothercustomer purchased 5 chocolates and 5 lollipops for Rs 10. Represent the situation algebraically and graphically. Solution: Step 1: Let the price of chocolate be Rsx , and that of lollipop be Rs y . Given, 4 chocolates and 5 lollipops were purchased for Rs 8. Step2: 8 5 4 ? ? ? y x Given, 5 chocolates and 5 lollipops were purchased for Rs 10. Step3: 10 5 5 ? ? ? y x Then the algebraic representation of the given situation is ) .( .......... .......... 8 5 4 I y x ? ? PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Page 4 PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now, I shall be two times as old as you will be”. Represent this situation algebraically and graphically. Solution: Step1: Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. Step2: Five years ago Maria’s age was(x – 5) years Her Son’s age was (y-5) years Step3: Two years later Maria’s age will be(x + 2) years Her son’s age will be (y + 2) years Step4: Given, 5 years ago Maria’s age = 2 times her son’s age then ? ) 5 ( 2 5 ? ? ? y x ? 0 5 2 ? ? ? y x Step5: Given, 2 years hence, Maria’s age = 2 times her son’s age then, ? ) 2 ( 2 2 ? ? ? y x ? 0 2 2 ? ? ? y x Thus, algebraic representation of the given situations is 0 5 2 ? ? ? y x ………. (I) 0 2 2 ? ? ? y x ………… (II) Step6: To obtain equivalent graphical representation we find two points on the line representing each equation. That is we find two solutions of each equation. Now, 2 2 ? ? y x ……………… FROM (II) ? 2 2 ? ? x y Therefore, 0 ? x ? 1 2 2 0 ? ? ? ? y 2 ? x ? 0 2 2 2 ? ? ? y Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step7: Plot the points A (0,-1) and B (2, 0) on a graph paper. Draw a line passing through the points A and B. Then, the line AB represents the equation 0 2 2 ? ? ? y x Step8: Also, consider 0 5 2 ? ? ? y x ? 2 5 ? ? x y Therefore, 1 ? x ? 3 2 5 1 ? ? ? y 3 ? x ? 4 2 5 3 ? ? ? y Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: x 1 3 y 3 4 Step9: Plot the points P(1,3) and Q(3,4) on the same graph paper. Draw a line passing through the point P and Q. Then the PQ represents the equation 0 5 2 ? ? ? y x We observe that line do not intersect anywhere. x 0 2 y -1 0 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 2. A shopkeeper sold 4 chocolates and 5 lollipops for Rs 8 to a customer. A little lateranothercustomer purchased 5 chocolates and 5 lollipops for Rs 10. Represent the situation algebraically and graphically. Solution: Step 1: Let the price of chocolate be Rsx , and that of lollipop be Rs y . Given, 4 chocolates and 5 lollipops were purchased for Rs 8. Step2: 8 5 4 ? ? ? y x Given, 5 chocolates and 5 lollipops were purchased for Rs 10. Step3: 10 5 5 ? ? ? y x Then the algebraic representation of the given situation is ) .( .......... .......... 8 5 4 I y x ? ? PAIR OF LINEAR EQUATIONS IN TWO VARIABLES ) ( .......... .......... 10 5 5 II y x ? ? Geometric/ graphical representation Step4: To obtain the equivalent graphical representation, we find two points on the line representing each equation, that is, we find two solution of each equation. Now, ) .( .......... .......... 8 5 4 FromI y x ? ? 5 4 8 x y ? ? ? Therefore, 2 ? x 0 5 2 4 8 ? ? ? ? ? y 2 ? ? x 2 . 3 5 16 5 )] 2 ( 4 [ 8 ? ? ? ? ? ? ? y Thus, the two solutions of the equation 8 5 4 ? ? y x are: x 2 -2 y 0 3.2 Step5: Plot the point A (2, 0) and B (-2, 3.2) on a graph paper. Draw a line passing through the points A and B. Then, the line AB represents the equation 8 5 4 ? ? y x . Step6: Now, ) ( .... .......... 10 5 5 II From y x ? ? 5 5 10 x y ? ? ? Therefore, 2 ? x 0 5 2 5 10 ? ? ? ? ? y 2 ? ? x 4 5 20 5 )] 2 ( 5 [ 10 ? ? ? ? ? ? ? y Thus, the two solutions of the equation 10 5 5 ? ? y x are: x 2 -2 y 0 4 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Page 5 PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now, I shall be two times as old as you will be”. Represent this situation algebraically and graphically. Solution: Step1: Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. Step2: Five years ago Maria’s age was(x – 5) years Her Son’s age was (y-5) years Step3: Two years later Maria’s age will be(x + 2) years Her son’s age will be (y + 2) years Step4: Given, 5 years ago Maria’s age = 2 times her son’s age then ? ) 5 ( 2 5 ? ? ? y x ? 0 5 2 ? ? ? y x Step5: Given, 2 years hence, Maria’s age = 2 times her son’s age then, ? ) 2 ( 2 2 ? ? ? y x ? 0 2 2 ? ? ? y x Thus, algebraic representation of the given situations is 0 5 2 ? ? ? y x ………. (I) 0 2 2 ? ? ? y x ………… (II) Step6: To obtain equivalent graphical representation we find two points on the line representing each equation. That is we find two solutions of each equation. Now, 2 2 ? ? y x ……………… FROM (II) ? 2 2 ? ? x y Therefore, 0 ? x ? 1 2 2 0 ? ? ? ? y 2 ? x ? 0 2 2 2 ? ? ? y Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step7: Plot the points A (0,-1) and B (2, 0) on a graph paper. Draw a line passing through the points A and B. Then, the line AB represents the equation 0 2 2 ? ? ? y x Step8: Also, consider 0 5 2 ? ? ? y x ? 2 5 ? ? x y Therefore, 1 ? x ? 3 2 5 1 ? ? ? y 3 ? x ? 4 2 5 3 ? ? ? y Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: x 1 3 y 3 4 Step9: Plot the points P(1,3) and Q(3,4) on the same graph paper. Draw a line passing through the point P and Q. Then the PQ represents the equation 0 5 2 ? ? ? y x We observe that line do not intersect anywhere. x 0 2 y -1 0 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 2. A shopkeeper sold 4 chocolates and 5 lollipops for Rs 8 to a customer. A little lateranothercustomer purchased 5 chocolates and 5 lollipops for Rs 10. Represent the situation algebraically and graphically. Solution: Step 1: Let the price of chocolate be Rsx , and that of lollipop be Rs y . Given, 4 chocolates and 5 lollipops were purchased for Rs 8. Step2: 8 5 4 ? ? ? y x Given, 5 chocolates and 5 lollipops were purchased for Rs 10. Step3: 10 5 5 ? ? ? y x Then the algebraic representation of the given situation is ) .( .......... .......... 8 5 4 I y x ? ? PAIR OF LINEAR EQUATIONS IN TWO VARIABLES ) ( .......... .......... 10 5 5 II y x ? ? Geometric/ graphical representation Step4: To obtain the equivalent graphical representation, we find two points on the line representing each equation, that is, we find two solution of each equation. Now, ) .( .......... .......... 8 5 4 FromI y x ? ? 5 4 8 x y ? ? ? Therefore, 2 ? x 0 5 2 4 8 ? ? ? ? ? y 2 ? ? x 2 . 3 5 16 5 )] 2 ( 4 [ 8 ? ? ? ? ? ? ? y Thus, the two solutions of the equation 8 5 4 ? ? y x are: x 2 -2 y 0 3.2 Step5: Plot the point A (2, 0) and B (-2, 3.2) on a graph paper. Draw a line passing through the points A and B. Then, the line AB represents the equation 8 5 4 ? ? y x . Step6: Now, ) ( .... .......... 10 5 5 II From y x ? ? 5 5 10 x y ? ? ? Therefore, 2 ? x 0 5 2 5 10 ? ? ? ? ? y 2 ? ? x 4 5 20 5 )] 2 ( 5 [ 10 ? ? ? ? ? ? ? y Thus, the two solutions of the equation 10 5 5 ? ? y x are: x 2 -2 y 0 4 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step7: Plot the point P (2, 0) and Q (-2, 4), on the same graph. Draw a line passing through the points P and Q. Then, the line PQ represents the equation 10 5 5 ? ? y x From the graph we observe that, the two lines representing the situation graphically, intersect at(2,0) GRAPHICAL METHOD OF SOLVING LINEAR EQUATIONS 3) 9 students of Class 8 took part in a Science Quiz. If the number of boys is 5 more than the number of girls, find the number of boys and girls who took part in the quiz. Solution: X axis- 1cm = 1 unit Y axis- 1cm = 1unit PAIR OF LINEAR EQUATIONS IN TWO VARIABLESRead More

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