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• Linear equations have a variable with a highest power of 1.
• A linear equation in one variable contains only one variable, also with a highest power of 1.
• Example: x + 1 = 3 is a linear equation.
• Example: x + x² + 2 = 0 is not a linear equation because the variable has a power of 2.
• 2x + 1 is a linear equation in one variable.
• x + y = 1 is not a linear equation in one variable because it contains two variables.
• Linear equations have two sides: LHS (left hand side) and RHS (right hand side).
• The "root" of the equation is when the LHS equals the RHS for a certain value of the variable.

Solving Linear Equations with Variable on One Side

• To solve for x in the equation 3x = 18:

• Divide both sides by 3: 3x/3 = 18/3

• This simplifies to x = 6.

• To solve for y in the equation 3y - 7 = 5:

• Add 7 to both sides: 3y - 7 + 7 = 5 + 7

• This simplifies to 3y = 12

• Divide both sides by 3: 3y/3 = 12/3

• Therefore, y = 4

• To solve for x when x/5 - 6/3 = 2:

• Find the least common multiple (LCM) of 5 and 3 which is 15.

• Rewrite the fractions with the common denominator: (3x - 30) / 15 = 2

• Multiply both sides by 15: [(3x - 30) / 15] * 15 = 2 * 15

• Simplify: 3x - 30 = 30

• Add 30 to both sides: 3x - 30 + 30 = 30 + 30

• Simplify: 3x = 60

• Divide both sides by 3: 3x / 3 = 60 / 3

• Therefore, x = 20

• To solve for m when 1/4 - m = -3/2:

• Find the LCM of 4 and 1, which is 4.

• Rewrite the equation: (1 - 4m) / 4 = -3/2

• Cross multiply: (1 - 4m) * 2 = -3 * 4

• Simplify: 2 - 8m = -12

• Subtract 2 from both sides: 2 - 8m - 2 = -12 - 2

• Simplify: -8m = -14

• Divide both sides by -8: m = -14 / -8

• Therefore, m = 7/4

Examples continued..

• To determine what number should be added to 1/2 to make 6:

• Set up the equation: 1/2 + x = 6

• Subtract 1/2 from both sides: x = 6 - 1/2

• Find a common denominator (2): x = 12/2 - 1/2

• Simplify: x = 11/2

• Therefore, 11/2 should be added to 1/2 to get 6

• To find four numbers when the sum of four multiples of 8 is 208:

• Define the numbers as x, x + 8, x + 16, and x + 24.

• The equation becomes: x + x + 8 + x + 16 + x + 24 = 208

• Combine like terms: 4x + 48 = 208

• Subtract 48 from both sides: 4x = 160

• Divide both sides by 4: x = 40

• The numbers are 40, 48, 56, and 64

• To find the breadth of a rectangle when the perimeter is 16/3 m and the length is 5/2 m:

• Set up the equation for the perimeter: 2(5/2 + x) = 16/3

• Simplify: 5 + 2x = 16/3

• Subtract 5 from both sides: 2x = 16/3 - 5

• Convert 5 to a fraction with a denominator of 3: 2x = 16/3 - 15/3

• Simplify: 2x = 1/3

• Divide both sides by 2: x = 1/6

• Therefore, the breadth of the rectangle is 1/6 m

• Mayank's mother is four times his age and after 7 years, the sum of their ages will be 64. To find their current ages:

• Let Mayank's age be x and his mother's age be 4x

• After 7 years, Mayank will be x + 7 and his mother 4x + 7

• The equation is: (x + 7) + (4x + 7) = 64

• Combine: 5x + 14 = 64

• Subtract 14: 5x = 50

• Divide by 5: x = 10, so Mayank is 10 and his mother is 40

Solving Linear Equations When Variables are on Both Sides

• When variables are on both sides of the equation, rearrange to isolate the variable.

• Solve 3x + 5 = 2x + 10

• Subtract 5 from both sides: 3x = 2x + 5

• Transpose 2x to the left: x = 5

• For x = 2/5 - x = 7/5, find x:

• Transpose x to the left: x = -7/5

• Isolate x terms on one side by finding common denominator: x = (-2/5) - (7/5)

• Combine: x = (-7/5)

• Divide by 3: x = (-7/3)

• To solve 2(4 – 5x) - 5 = (3x + 1):

• Distribute and simplify: 8 - 10x - 5 = 3x + 1

• Combine constants: 3 - 10x = 3x + 1

• Transpose terms: -10x - 3x = 1 - 3

• Simplify: -13x = -2

• Divide: x = 2/13

• Given 3.9m – 3(1.5 – m) = 1.5m – 1.8, find m:

• Distribute and simplify: 3.9m - 4.5 + 3m = 1.5m - 1.8

• Combine like terms: 6.9m - 4.5 = 1.5m - 1.8

• Transpose terms: 6.9m - 1.5m = 4.5 - 1.8

• Simplify: 5.4m = 2.7

• Divide: m = 1/2

Example 13

• If the sum of a two-digit number and its digits interchanged is 165 and digfference in the digits id 3, find the number
• Express this as another digit will be x-3, for the values 10(x-3)+x and 10x + (x - 3)
• It is possible to also say they are [10(x − 3) + x] + [10x + (x − 3)] = 165. giving the values -10x − 30 + x + 10x + x - 3 = 165
• 22x-33 = 165
• 22x = 165 + 33
• 22x = 198 / 22
• x = 9 resulting in an other digit of 9 - 3 = 6
• So, the resulting number is 69.

Example 14

Akshit will be four times his current age, 15s years after.

• We can express this as what his age is now (x) to his age after 15(x+15 = 4c) making his age x=4x-15
• We then simplyfiy 15 = 3x to get to x = 5