Class 8 Maths Linear Equations in One Variable

Linear Equations in One Variable – Quick Notes

Definition:

A linear equation in one variable is an equation that can be written in the form:$$ax + b = 0$$ax+b=0

Where:

• $x$x = variable
• $a$a and $b$b = constants ($a \neq 0$a=0)

Steps to Solve:

1. Simplify both sides (remove brackets).
2. Bring variable terms to one side and constants to the other.
3. Solve for the variable by dividing/multiplying as needed.

Example:
Solve $3x + 5 = 11$3x+5=11$$3x = 11 – 5 = 6 \implies x = 2$$3x=11−5=6⟹x=2

Properties:

1. A linear equation has exactly one solution.
2. Can have infinite solutions (e.g., $0x = 0$0x=0)
3. Can have no solution (e.g., $0x = 5$0x=5)

Linear Equations in One Variable – Q&A (Class 8 NCERT)

1. Q: Which of the following is a linear equation in one variable?
   A: $2x + 5 = 11$2x+5=11 ✅
2. Q: Solve for $x$x: $2x + 5 = 11$2x+5=11
   A: $2x = 6 \implies x = 3$2x=6⟹x=3
3. Q: Solve for $x$x: $3x – 7 = 2$3x−7=2
   A: $3x = 9 \implies x = 3$3x=9⟹x=3
4. Q: An equation has infinite solutions. Which type is it?
   A: $0x = 0$0x=0
5. Q: An equation has no solution. Which type is it?
   A: $0x = 5$0x=5
6. Q: Solve: $4x + 3 = 19$4x+3=19
   A: $4x = 16 \implies x = 4$4x=16⟹x=4
7. Q: Which of the following is not a linear equation in one variable?
   A: $x^2 + 5 = 0$x2+5=0
8. Q: Solve: $-5x + 10 = 0$−5x+10=0
   A: $-5x = -10 \implies x = 2$−5x=−10⟹x=2
9. Q: Solve: $\frac{x}{2} + 3 = 7$2x​+3=7
   A: $\frac{x}{2} = 4 \implies x = 8$2x​=4⟹x=8
10. Q: Solve: $7 – 3x = 1$7−3x=1
    A: $-3x = -6 \implies x = 2$−3x=−6⟹x=2
11. Q: Which property ensures $ax + b = 0$ax+b=0 has exactly one solution if $a \neq 0$a=0?
    A: The coefficient of $x$x ($a \neq 0$a=0) ensures exactly one solution.
12. Q: Solve for $x$x: $\frac{x}{3} + 5 = 10$3x​+5=10
    A: $\frac{x}{3} = 5 \implies x = 15$3x​=5⟹x=15
13. Q: Equation $0x = 0$0x=0 has how many solutions?
    A: Infinite solutions ✅
14. Q: Equation $0x = 5$0x=5 has how many solutions?
    A: No solution ❌
15. Q: Solve: $2(x – 3) = 4$2(x−3)=4
    A: $x – 3 = 2 \implies x = 5$x−3=2⟹x=5
16. Q: Solve: $3(x + 5) – 2 = 13$3(x+5)−2=13
    A: $3x + 15 – 2 = 13 \implies 3x + 13 = 13 \implies 3x = 0 \implies x = 0$3x+15−2=13⟹3x+13=13⟹3x=0⟹x=0
17. Q: Which equation has $x = 4$x=4 as a solution?
    A: $x – 4 = 0$x−4=0 ✅
18. Q: If $2x + 3 = 2x + 7$2x+3=2x+7, how many solutions exist?
    A: No solution ❌
19. Q: Solve: $5x – 2x + 1 = 10$5x−2x+1=10
    A: $3x + 1 = 10 \implies 3x = 9 \implies x = 3$3x+1=10⟹3x=9⟹x=3
20. Q: Translate: “The sum of a number and 7 is 12.”
    A: Let the number = $x$x; $x + 7 = 12 \implies x = 5$x+7=12⟹x=5