∴ x = 5 .... (Dividing both the sides by 13) Substituting x = 5 in equation (1), 3x + 2y = 29 3 × [5] + 2y = 29 [15] + 2y = 29 2y = 29 – [15] ∴ 2y = [14] ∴ y = 7 (x, y) = ([5],[7]) is the solution Graph of 3 Linear Equation in Two Variables :
Steps to follow for drawing a Graph of Linear Equation in two variables :
Activity (I) (Text Book) : Q. Solve the following simultaneous equations by graphical method.
The coordinates of the point of intersection are ( – 1, – 2). (x, y) = ( – 1, – 2) is the solution. Activity II (Text Book) : Solve the above equations by method of elimination. Check your solution with the solution obtained by graphical method. x – y =1 ……(1) 5x – 3y = 1 …….(2) Multiplying equation (1) by 3, 3x – 3y = 3 ……..(3) Subtracting equation (3) from equation (2), \(\begin{array}{rrrrrrr} & 5x & - & 3y & = & 1 &\text{…..(2)}\\ - & & & & & & \\ &3x & - & 3y & = & 3 &\text{…..(3)} \\ -& & +& & &-& \\ \hline & 2x & & & = & -2 & \\ \end{array}\) ∴ x = –1 ….(Dividing both the sides by 2) Substituting x = -1 in equation (1), –1 – y = 1 –y = 1 + 1; ∴ –y = 2 ∴ y = –2 (–1, –2) is the solution of the given equations The solution in both the methods is the same.
Let’s discuss. (Text Book) To solve simultaneous equations x + 2y = 4 ; 3x + 6y = 12 graphically, following are the ordered pairs. x + 2y = 4
3x + 6y = 12
Plotting the above ordered pairs, graph is drawn. Observe it and find answers of the following questions. (1) Are the graphs of both the equations different or same ? (2) What are the solutions of the two equations x + 2y = 4 and 3x + 6y = 12 ? How many solutions are possible ? (3) What are the relations between coefficients of x, coefficients of y and constant terms in both the equations ? (4) What conclusion can you draw when two equations are given but the graph is only one line ? (1) The graphs of both the equations are same. (2) The solutions of the given equations are ( –4, 4), ( –2, 3), (0, 2), (1, 1.5), (2, 1), (8, –2). Infinite solutions are possible. (3) The ratio of the coefficients of x = \(\frac {a_1}{a_2}\) = \(\frac 13\) ….(1) The ratio of the coefficients of y = \(\frac {b_1}{b_2}\) = \(\frac 26\) = \(\frac 13\) …..(2) The ratio of the constants = \(\frac {c_1}{c_2}\) = \(\frac{4}{12}\) = \(\frac 13\) ….(3) All these ratios are the same. (4) If a1x + b1y = c1 and a2x + b2y = c2 are the given equations and if \(\frac {a_1}{a_2}\) = \(\frac {b_1}{b_2}\) = \(\frac {c_1}{c_2}\) , then the graph of the given two equations is only one line, having infinite solutions. Let us consider another example (Text Book) : Draw graphs of x - 2y = 4, 2x - 4y = 12 on the same co-ordinate plane. Observe it. Think of the relation between the coefficients of x, coefficients of y and the constant terms and draw the inference. x - 2y = 4 \ y = x-4/2
2x - 4y = 12 \ y= 2x-12/4
The graphs of the given equations are parallel lines. The ratio of the coefficients of
Considering the given equations as a1x + b1y = c1 and a2x + b2y = c2, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}\) Inference : If \(\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}\) then the graphs of the given two equations are parallel lines having no solution. Determinant : \(\begin{vmatrix} a &b \\ c& d \end{vmatrix}\) is a determinant of four numbers a, b, c, d. (a, b), (c, d) are rows and \(\begin{pmatrix} a\\c \end{pmatrix},\begin{pmatrix} b\\d \end{pmatrix}\) are columns. The degree of this determinant is 2, because there are 2 elements in each column and 2 elements in each row. The determinant represent a number which is (ad-bc). = (ad-bc). ad - bc is the value of the determinant \(\begin{vmatrix} a &b \\ c& d \end{vmatrix}\) Determinants are usually represented with capital letters A, B, C, etc. Determinant Method (Cramer’s Rule) : Determinant method of solving simultaneous equations was first given by a Swiss mathematician Gabriel Cramer. So, it is also known as Cramer’s rule. To use Cramer’s rule, the equations are written as a1x + b1y = c1 and a2x + b2y = c2 Coefficient of x is a1 and a2, coefficient of y is b1 and b2, coefficient of constant terms is c1 and c2 then D = \(\begin{vmatrix} a_1 &b_1 \\ a_2& b_2 \end{vmatrix}\) - In D the column of constants \(\begin{pmatrix} c_1\\c_2 \end{pmatrix}\) is omitted Dx = \(\begin{vmatrix} c_1 &b_1 \\ c_2& b_2 \end{vmatrix}\) - In Dx the column of the coefficients of x, \(\begin{pmatrix} a_1\\a_2 \end{pmatrix}\) is replaced by \(\begin{pmatrix} c_1\\c_2 \end{pmatrix}\) Dy = \(\begin{vmatrix} a_1 &c_1 \\ a_2& c_2 \end{vmatrix}\)- In Dy the column of the coefficients of y, \(\begin{pmatrix} b_1\\b_2 \end{pmatrix}\) is replaced by \(\begin{pmatrix} c_1\\c_2 \end{pmatrix}\) \(x=\frac{D_x}{D}\) and \(y=\frac{D_y}{D}\) Cramer’s method to solve simultaneous equations : Write given equations in the form ax + by = c . Find the values of determinants D using D = \(\begin{vmatrix} a_1 &b_1 \\ a_2& b_2 \end{vmatrix}\) Find the values of determinants Dx using Dx = \(\begin{vmatrix} c_1 &b_1 \\ c_2& b_2 \end{vmatrix}\) and Find the values of determinants Dy using Dy = \(\begin{vmatrix} a_1 &c_1 \\ a_2& c_2 \end{vmatrix}\) Using, \(x=\frac{D_x}{D}\) and \(y=\frac{D_y}{D}\) find values of x, y. Activity : (Textbook page) To solve the simultaneous equations by determinant method, fill in the blanks : y + 2x – 19 = 0; 2x – 3y + 3 = 0. Solution : Write the given equations in the form ax + by = c. 2x + y = 19 2x – 3y = –3 D = \(\begin{vmatrix} 2 & 1 \\ 2& -3 \end{vmatrix}\) = ([2]x-3)-(2x[1]) = [-6] – [2] = [–8] Dx = \(\begin{vmatrix} 19 & 1 \\ -3& -3 \end{vmatrix}\) = (19 x[-3])-([-3]x[1]) = [–57] – [–3] = [–54] Dy = \(\begin{vmatrix} 2 & 19 \\ 2& -3 \end{vmatrix}\) = ([2]x[-3])-([2]x[19]) = [–6] – [38] = [–44] By Cramer’s Rule – \(x=\frac{D_x}{D} = \frac{-54}{-8} = \frac{27}{4}\) and \(y=\frac{D_y}{D}=\frac{-44}{-8}=\frac{11}{2}\) ∴ \((x,y)=(\frac{27}{4},\frac{11}{2})\) is the solution of the given equation Activity 2 : Complete the following activity – 3x – 2y = 3, 2x + y = 16 Find the values of determinants in the given equations D = \(\begin{vmatrix} 3 & -2 \\ 2& 1 \end{vmatrix}\) = (3x1)–(2x–2) = [7] Dx = \(\begin{vmatrix} 3 & -2 \\ 16& 1 \end{vmatrix}\) = (3x1)–(16 x –2) = [35] Dy = \(\begin{vmatrix} 3 & 3 \\ 2&16 \end{vmatrix}\) = (3x16)–(2 x 3) = [42] Values according to Cramer’s Rule \(x=\frac{D_x}{D} = \frac{35}{7}\) = [5] \(y=\frac{D_y}{D}=\frac{42}{7}\) = [6] ∴ (x,y) = ([5], [6]) is the solution. Let’s think (Textbook) Q What is the nature of the solution if D = 0? D = \(\begin{vmatrix} a_1 & b_1 \\ a_2& b_2 \end{vmatrix}\) If D = 0, then a1b2 – b1a2 = 0. Or \(\frac{a_1}{a_2}= \frac{b_1}{b_2}\) ∴ The two simultaneous equations have either no solution or infinite solutions. It means the two simultaneous equations do not have a unique solution. Q What can you say about lines if common solution is not possible? If the common solution about the lines is not possible, then the lines are parallel to each other or coincide. Equations reducible to a pair of linear equations in two variables : Activity : Complete the following table.
In the above table the equations are not linear. We can create new variables making a proper change in the given variables. Substituting the new variables in the given non-linear equations, we can convert them in linear equations. Remember that the denominator of any fraction of the form \(\frac mn\) cannot be zero. Let’s Study the following example : \(\frac{4}{x}+\frac{5}{y}\)= 7; \(\frac{3}{x}+\frac{4}{y}\)= 5 \(4(\frac{1}{x})+5(\frac{1}{y})\)= 7 .....(1) \(3(\frac{1}{x})+4(\frac{1}{y})\)= 5 ....(2) Substituting a for \(\frac{1}{x}\) and b for \(\frac{1}{y}\) , 4a + 5b = 7 …..(3) 3a + 4b = 5 …..(4) These are linear equations in variables a and b. Multiplying both the sides of equation (3) by 4 and equation (4) by 5, And subtracting equation (6) from equation (5) \(\begin{array}{rrrrrrr} & 16a & + & 20b & = & 28 &\text{…..(2)}\\ - & & & & & & \\ &15a & + & 20b & = & 25 &\text{…..(3)} \\ -& & -& & &-& \\ \hline & a & & & = & 3 & \\ \end{array}\) Substituting a = 3 in equation (3), 4(3) + 5b = 7 ∴ 12 + 5b = 7 ∴ 5b =7 – 12; \ 5b = –5 b = –1. Now, a = \(\frac{1}{x}\) = 3 ∴ x = \(\frac{1}{3}\); b = \(\frac{1}{y}\) = –1 ∴ y = –1 Answer- (x, y) = \((\frac{1}{3},-1)\) is the solution of the given simultaneous equations. Activity : To solve given equations fill the boxes below suitably. \(\frac{5}{x-1}+\frac{1}{y-2}\)= 2; \(\frac{6}{x-1}-\frac{3}{y-2}\)= 1 Replacing \(\frac{1}{x-1}\) by m \(\frac{1}{y-2}\) by n ∴ 5m + n = 2; 6m - n = 1 New equations on solving : Multiplying (5m + n = 2) by 3 ∴ 15m + 3n = 6 \(\begin{array}{rrrrrrr} & 15m & + & 3n & = & 6 &\text{…..(2)}\\ + & & & & & & \\ &6m & - & 3n & = & 1 &\text{…..(3)} \\ & & & & && \\ \hline & 21m & & & = & 7 & \\ \end{array}\) ∴ m = \(\frac{7}{21}\) ∴ m = = \(\frac{1}{3}\) 15m + 3n = 6 ∴ 15 \(\frac{1}{3}\) + 3n = 6 ∴ 5 + 3n = 6, ∴ 3n = 6-5 = 1 ∴ n = \(\frac{1}{3}\) m = \(\frac{1}{3}\) ; n = \(\frac{1}{3}\) Replacing m,n by their original values \(\frac{1}{x-1}\) = \(\frac{1}{3}\) \(\frac{1}{y-2}\) = \(\frac{1}{3}\) On solving x – 1 = 3, ∴ x = 3 + 1 = 4 y – 2 = 3, ∴ y = 3 + 2 = 5 ∴ x = 4, y = 5 Answer- (x, y) = (4,5) is the solution of the given simultaneous equations. Application of Simultaneous Equations : Use of simultaneous equations is common in Science and Mathematics. The simultaneous equations are used to solve word problems. To solve the problem, apply the following steps :
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