Title Chapter - 4 Linear_Equations_in_Two_Variables.pdf ✅

ChemContent Linear Equations in Two Variables 1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. 2. Linear equations in one variable, of the type ax + b = 0, can also expressed as a linear equation in two variables. Since, ax + b = 0 ⇒ ax + 0.y + b =0. 3. A solution of a linear equation in two variables is a pair of values, one for x and one for y, which satisfy the equation. 4. The solution of a linear equation is not affected when- i. The same number is added or subtracted from both the sides of an equation. ii. Multiplying or dividing both the sides of the equation by the same non-zero number. 5. A linear equation in two variables has infinitely many solutions. 6. Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line. 7. A linear equation in two variables is represented geometrically by a straight line whose points make up the collection of solutions of the equation. This is called the graph of the linear equation. 8. x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis. 9. The graph of x = k is a straight line parallel to the y-axis. For example, the graph of the equation x = 5 is as follows: 10. The graph of y = k is a straight line parallel to the x-axis.
ChemContent For example, the graph of the equation y = 5 is as follows: 11. An equation of the type y = mx represents a line passing through the origin, where m is a real number. For example, the graph of the equation y = 2x is as follows: Linear equation in one variable When an equation has only one variable of degree one, then that equation is known as linear equation in one variable. A linear equation in one variable is an equation which has a maximum of one variable of order 1. It is of the form ax + b = 0, where x is the variable. • Standard form: ax + b = 0, where a and b ε R & a ≠ 0 • Examples of linear equation in one variable are:
ChemContent – 3x - 9 = 0 – 2t = 5 Standard Form of Linear Equations in One Variable The standard form of linear equations in one variable is represented as: ax + b = 0 Where, • ‘a’ and ‘b’ are real numbers. • Both ‘a’ and ‘b’ are not equal to zero. Thus, the formula of linear equation in one variable is ax + b = 0. Solving Linear Equations in One Variable For solving an equation having only one variable, the following steps are followed • Step 1: Using LCM, clear the fractions if any. • Step 2: Simplify both sides of the equation. • Step 3: Isolate the variable. • Step 4: Verify your answer. Example of Solution of Linear Equation in One Variable Let us understand the concept with the help of an example. For solving equations with variables on both sides, the following steps are followed: Consider the equation: 5x – 9 = -3x + 19 Step 1: Transpose all the variables on one side of the equation. By transpose, we mean to shift the variables from one side of the equation to the other side of the equation. In the method of transposition, the operation on the operand gets reversed. In the equation 5x – 9 = -3x + 19, we transpose -3x from the right-hand side to the left- hand side of the equality, the operation gets reversed upon transposition and the equation becomes: 5x – 9 +3x = 19 ⇒ 8x -9 = 19 Step 2: Similarly transpose all the constant terms on the other side of the equation as below: 8x -9 = 19 ⇒ 8x = 19 + 9 ⇒ 8x = 28 Step 3: Divide the equation with 8 on both sides of the equality. 8x/8 = 28/8
ChemContent ⇒ x = 28/8 If we substitute x = 28/8 in the equation 5x – 9 = -3x + 19, we will get 9 = 9, thereby satisfying the equality and giving us the required solution. The Application of Linear equation There are various applications of linear equations in Mathematics as well as in real life. An algebraic equation is an equality that includes variables and equal sign (=). A linear equation is an equation of degree one. The knowledge of mathematics is frequently applied through word problems, and the applications of linear equations are observed on a wide scale to solve such word problems. Here, we are going to discuss the linear equation applications and how to use them in the real world with the help of an example. A linear equation is an algebraic expression with a variable and equality sign (=), and whose highest degree is equal to 1. For example, 2x – 1 = 5 is a linear equation. • Linear equation with one variable and degree one is called a linear equation in one variable. (Eg, 3x + 5 = 0) • Linear equation with degree one and two variables is called a linear equation in two variables. (Eg, 3x + 5y = 0) The graphical representation of linear equation is ax + by + c = 0, where, • a and b are coefficients • x and y are variables • c is a constant term In real life, the applications of linear equations are vast. To tackle real-life problems using algebra, we convert the given situation into mathematical statements in such a way that it clearly illustrates the relationship between the unknowns (variables) and the information provided. The following steps are involved while restating a situation into a mathematical statement: • Translate the problem statement into a mathematical statement and set it up in the form of algebraic expression in a manner it illustrates the problem aptly. • Identify the unknowns in the problem and assign variables (quantity whose value can change depending upon the mathematical context) to these unknown quantities. • Read the problem thoroughly multiple times and cite the data, phrases and keywords. Organize the information obtained sequentially. • Frame an equation with the help of the algebraic expression and the data provided in the problem statement and solve it using systematic techniques of equation solving. • Retrace your solution to the problem statement and analyze if it suits the criterion of the problem.