Master Linear Equations in One Variable (Grade 08 CBSE)

  

Master Linear Equations in One Variable (Grade 08 CBSE)

Hey there, future math whizzes! Ready to unlock a fundamental skill that'll make algebra a breeze? Today, we're diving into the exciting world of Linear Equations in One Variable. Don't let the name intimidate you – by the time we're done, you'll be solving these equations like a pro!

Master Linear Equations in One Variable (Grade 08 CBSE)

What Exactly is a Linear Equation?

Imagine a perfectly balanced seesaw. An equation is just like that – a statement where two expressions are equal. In math, we use an equals sign (=) to show this balance.

A linear equation in one variable is simply an equation with only one unknown value (our "variable," usually represented by letters like x, y, or z), and this variable is never raised to a power higher than one. Think of it as a straight line if you were to graph it!


Here are some quick examples:

·        x+5=12

·        3y=15

·        z−7=2

·        2p+4=10

Our main goal? To find the specific value of that variable that makes the equation true. That's what we call the solution!

Solving Equations: Your Step-by-Step Guide

The secret to solving any linear equation is to keep the "seesaw" balanced. Whatever you do to one side of the equation, you must do to the other side. This helps us isolate the variable and find its value.

1. When the Variable is on One Side

These are the simplest types, like warm-up exercises! We use inverse operations to get our variable all by itself.

·        Addition & Subtraction:

o   Example 1: Getting rid of addition

x+7=15

To isolate x, we do the opposite of adding 7 – we subtract 7 from both sides:

x+7−7=15−7

x=8

o   Example 2: Tackling subtraction

y−3=10

The opposite of subtracting 3 is adding 3. So, add 3 to both sides:

y−3+3=10+3

y=13

·        Multiplication & Division:

o   Example 3: Undoing multiplication

4z=20

To free z, we divide both sides by 4 (the opposite of multiplying by 4):

44z​=420​

z=5

o   Example 4: Reversing division

2a​=6

The opposite of dividing by 2 is multiplying by 2. Multiply both sides by 2:

2a​×2=6×2

a=12

2. When the Variable is on Both Sides

This is where it gets a little more exciting! Your first move is to gather all the variable terms on one side and all the constant numbers on the other.

·        Example 5: Bringing variables together

5x−3=2x+9

1.     Let's bring all x terms to the left. Subtract 2x from both sides:

5x−3−2x=2x+9−2x

3x−3=9

2.     Now, let's move the constant terms to the right. Add 3 to both sides:

3x−3+3=9+3

3x=12

3.     Finally, divide both sides by 3 to find x:

33x​=312​

x=4


Master Linear Equations in One Variable -Simplifying Complex Equations

Sometimes, equations come disguised with parentheses or fractions. No worries! We just need to "reduce" them to a simpler form before we can solve.

1. Equations with Parentheses

Use the distributive property to get rid of those pesky parentheses.

·        Example 6: Distribute and solve

3(x+2)=18

1.     Distribute the 3 to everything inside the parenthesis:

3×x+3×2=18

3x+6=18

2.     Now it looks familiar! Subtract 6 from both sides:

3x+6−6=18−6

3x=12

3.     Divide by 3:

33x​=312​

x=4

2. Equations with Fractions

Don't let fractions scare you! We can eliminate them by multiplying every single term by the Least Common Multiple (LCM) of the denominators.

·        Example 7: Clearing fractions with LCM

3x​+2x​=5

1.     The LCM of 3 and 2 is 6. Multiply every single term by 6:

6×3x​+6×2x​=6×5

2x+3x=30

2.     Combine the x terms:

5x=30

3.     Divide by 5:

55x​=530​

x=6

The All-Important Step: Verifying Your Solution!

Found your solution? Great! But don't just stop there. Always, always verify it! Substitute your calculated value of the variable back into the original equation. If both sides match, you've nailed it!

Let's verify Example 5 (5x−3=2x+9), where we found x=4:

Left Hand Side (LHS): 5x−3=5(4)−3=20−3=17

Right Hand Side (RHS): 2x+9=2(4)+9=8+9=17

Since LHS = RHS, our solution x=4 is absolutely correct!

 Master Linear Equations in One Variable -Worksheet

Time to put your new skills to the test. Solve these linear equations and don't forget to verify your answers!

1.     m+9=21

2.     7p=42

3.     y−12=−5

4.     4a​=9

5.     6x+5=2x+17

6.     3(t−4)=15

7.     8−3k=2k+18

8.     5y​−2y​=3

9.     5(z+1)=2(z−2)+7

10.  3x+1​=4x−2​

 

Master Linear Equations in One Variable: Test Paper

Think you've got it mastered? Take this quick test to check your progress. Show all your steps!

Total Marks: 20

Section A: Multiple Choice Questions (5 marks)

1.     Which of the following is a linear equation in one variable?

(a) x2+2=7                                          (b) 2x+3y=5

(c) 4p−1=11                                       (d) x(x+1)=0

2.     The solution to the equation y−8=−3 is:

(a) 5                                                     (b) -5

(c) 11                                                   (d) -11

3.     If 3m=27, then m is:

(a) 81                                                   (b) 9

(c) 30                                                   (d) 24

4.     When solving 5x−7=3x+1, what's the smartest first step to bring all x terms to one side?

(a) Add 7 to both sides                       (b) Subtract 1 from both sides

(c) Subtract 3x from both sides          (d) Add 5x to both sides

5.     What's the LCM of the denominators in the equation 2x​+4x​=6?

(a) 2                                                     (b) 4

(c) 6                                                     (d) 8

Section B: Short Answer Questions (10 marks)

Solve the following equations:

6.     x+15=40 (2 marks)

7.     9a=72 (2 marks)

8.     7n−6=2n+24 (3 marks)

9.     2(p+3) =16 (3 marks)

Section C: Long Answer Questions (5 marks)

10.  Solve and verify your answer: 3y​−4y​=2 (5 marks)

 

Answer Key (Worksheet):

1.     m=12

2.     p=6

3.     y=7

4.     a=36

5.     x=3

6.     t=9

7.     k=−2

8.     y=−10

9.     z=−2

10.  x=−7

Answer Key (Test Paper):

Section A:

1.     c) 4p−1=11

2.     a) 5

3.     b) 9

4.     c) Subtract 3x from both sides

5.     b) 4

Section B:

6. x=25

7. a=8

8. n=6

9. p=5

Section C:

10. y=24

Verification:

LHS = 324​−424​=8−6=2

RHS = 2

Since LHS = RHS, the solution is correct!

Keep practicing these equations, and you'll find them incredibly useful in higher classes and even in everyday problem-solving. How did you do on the worksheet and test? Let us know in the comments below!

 

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