Linear Equation in Two Variables

• Methods of solving linear equations in two variables include the graphical method and Cramer’s method.
• Equations that can be transformed into a linear equation in two variables are also known as simultaneous equations.
• Determinants are usually represented with capital letters as A, B, C, D, etc.
• In Cramer’s method, x and y are variables, a 1, b 1, c 1 and a 2, b 2, c 2 are real numbers, and a 1 b 2 - a 2 b 1 is zero.
• The degree of a determinant is 2, as there are 2 elements in each column and 2 elements in each row.
• To use Cramer’s method, the equations are written as a 1 x + b 1 y = c 1 and a 2 x + b 2 y = c 2.
• The determinant represents a number which is (ad-bc), for example, in the case of A = 5, 3, 7, 9.
• The digit in units place in a number is equal to the sum of the digits in the number.
• The digit in hundreds place in a number is equal to the sum of the digits in the number minus 1.
• The number obtained by reversing the digits in a number is 100 y + 10(x + y + 1) + x.
• The sum of the digits in a three digit number is 100 x + 10(x + y + 1) + y.
• The number 153 is obtained by reversing the digits in the number 110 x + 11 y + 10.
• The digit in tens place in a number is equal to the sum of the digits in the number minus 2.
• The sum of the digits in a given number is 2 x + 2 y + 1.
• Given number + 198 = new number.
• Gabriel Cramer was very well versed in mathematics from childhood.
• In D, the column of constants c is omitted.
• To solve simultaneous equations using Cramer’s Rule, find the values of D, Dx, and Dy.
• In Dx, the column of the coefficients of x, a12, is replaced by c.
• Gabriel Cramer was a professor in Geneva.
• Gabriel Cramer, a Swiss mathematician, was born in Geneva on July 31, 1704 and died on January 4, 1752.
• In Dy, the column of the coefficients of y, b12, is replaced by c.
• At the age of eighteen, Gabriel Cramer obtained a doctorate.
• The equation ax ± by = c can be rewritten as Dx ± Dy = C, where D is a determinant, x and y are unknowns, and C is a constant.
• To draw a graph of a linear equation in two variables, find at least 4 ordered pairs for the given equation, draw X-axis and Y-axis on graph paper, plot the points, and observe that all 4 points lie on one line.
• The graphical method involves drawing graphs of x + y = 4 and 2 x - y = 2, observing that the two lines intersect at (2, 2), and noting that the point (2, 2) satisfies both equations.
• The solution of these equations can also be obtained by method of elimination.
• The values of variables that satisfy the given equations, give the solution of the given equations.
• The method of elimination involves solving x + y = 4 and 2 x - y = 2 separately, and then substituting the value x = 2 in one of the equations to get y = 2.
• The solution of the given equations x + y = 4 and 2 x - y = 2 is x = 2, y = 2.
• After solving these equations, the solution is found by dividing both sides of the equation by 32, resulting in x + y = 1.
• 5 x - 6 + 9 x = 8 can be rewritten as 14 x - 6 = 8.
• 14 x = 8 + 6 can be rewritten as 14 x = 14.
• Subtracting equation (II) from (I) results in - x + y = 5.
• x = 1 in equation (III) results in y = -1.
• In the two equations 15 x + 17 y = 21; 17 x + 15 y = 11, the coefficients of x and y are interchanged.
• Solving 3 x + 2 y = 29; 5 x - y = 18 results in y = ( x , y ) = ( -1, -1 ).
• Adding equations (III) and (V) results in x + y = 1 - x + y = 5.
• Placing the value 3 in equation (III) results in y = 3.
• 5 x - 3 y = 8 can be rewritten as 5 x - 3(2 - 3 x) = 8.