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!
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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