2Linear Equation in One Variable A linear equation in one variable is an equation that can be written in the formax + b = 0Where a 0For example:5x – 4 = 0, 2x+1=0, 8x + 9 = 2
3PROPERTIES OF AN EQUATION If same quantity is added to both sides of the equation, the sums are equal.Thus: x=7 => x + a=7+aIf same quantity is subtracted from both sides of an equation, the differences are equalThus: x=7 => x-a=7-aIf both the sides of an equation are multiplied by the same quantity, the products are equal.Thus: x=7 => ax=7aIf both the sides of an equation are divided by the same quantity, the quotients are equal.Thus: x=7 => x ÷ a=7÷a
4Solving EquationsTo solve an equation means to find all values that make the equation a true statement. Such values are called solutions, and the set of all solutions is called the solution set.
5Solving equations graphically Graph both sides of the equation and see where they intersect!They intersect at (-2, -20). Since the x value of this coordinate is -2, then the solution to the equation is -2.
6Steps to Solve a Linear Equation in One Variable Simplify both sides of the equation.• Clear parentheses.• Consider clearing fractions or decimals (if any are present) by multiplying both sides of the equation by a common denominator of all terms.• Combine like terms.Use the addition or subtraction property of equality to collect the variable terms on one side of the equation.Use the addition or subtraction property of equality to collect the constant terms on the other side of the equation.Use the multiplication or division property of equality to make the coefficient of the variable term equal to 1.Check your answer.
8* Not linear expression : 1 + z + z² 2x,2x +1,3y-7,12-5z,5/4(x-4)+10* Not linear expression :1 + z + z²An algebraic equation is an equality involving variables. It has a equality sign. The expression on the left of the equality sign is the left hand side (LHS). The expression on the right of the equality sign is the right hand side (RHS).In an equation on the values of the expression on the LHS and RHS are equal. This happens to be true only for certain values are the solution of the equation.
9TO SOLVE AN EQUATION 1.To solve an equation of the form x + a=b E.g.: Solve x+4=10Solution: x+4= => x+4-4=10-4 (subtracting 4 from both the sides)=> x=62.To solve an equation of the form x-a=bE.g.: Solve y-6=5 equal.Solution: y-6= => y-6+6=5+6 (adding 6 to both sides)=> y=11
103.To solve an equation of the form ax=b E.g.: Solve 3x=9Solution: 3x= =>=> x = 34. To solve an equation of the form x/a=bE.g.: Solve = 6Solution: = => ×2=6×2=> x=12
11Conditional Equations The equation x+ 4 = 6 is a conditional equation because it is true on the condition that x = 2. For other values of x, the statement x + 4 = 6 is false.
12Contradiction Solve: x + 1 = x + 2 1 = 2 is a contradiction. This equation has no solution. And it is NOT linear.
13IdentitiesAn equation that has all real numbers as a solution is an Identity.For example,X + 4 = X + 44 = 4 is a true statement. Therefore, the solution is all real numbers.
14Important pointsAn algebraic is an equality variables. It says that the values of the expression on one side of the equality sign is equal to the value of the expression on the other side.A linear equation may have for its solution any rational number.The utility of linear equation is in their diverse applications ; different problems on numberThe utility of linear equation is their diverse applications ; different problems on numbers ,ages ,perimeters ,combination of currency notes , and so on can solved using linear equation.An equation may have linear expression on both sides.
15Work sheet Solve: x − 2 = 7 Ans. Transposing 2 to R.H.S, we obtain Solve : y + 3 = 10Ans. Transposing 3 to R.H.S, we obtainy = 10 − 3 = 7Solve : 14y − 8 = 13Ans. Transposing 8 to R.H.S, we obtain14y =14y = 21Dividing both sides by 14, we obtain
16Qn . If you subtract from a number and multiply the result by, you get. What is the number?Ans. Let the number be x. According to the question,On multiplying both sides by 2, we obtainOn transposing to R.H.S, we obtainTherefore, the number is .
17Solve the linear equation L.C.M. of the denominators, 3 and 5, is 15.Multiplying both sides by 15, we obtain5(x − 5) = 3(x − 3)⇒ 5x − 25 = 3x − 9 (Opening the brackets)⇒ 5x − 3x = 25 − 9⇒ 2x = 16