Chapter 4:Simple Equations
Overview of the Chapter
Introduction to Simple Equations
“Simple Equations” in Class 7 Mathematics introduces the basic concept of algebraic equations. This chapter covers the formulation, solving, and application of simple equations. Understanding these concepts is essential for progressing in algebra and solving real-world problems involving unknown values.
What is an Equation?
Definition and Examples
Definition: An equation is a mathematical statement that shows the equality of two expressions separated by an equals sign (=).
Example: (2x + 3 = 7) is an equation where (2x + 3) and (7) are expressions.
Formulating Simple Equations
Word Problems: Translate word problems into algebraic equations.
Example: “A number increased by 5 is 12” can be written as (x + 5 = 12).
Solving Simple Equations
Method of Balancing
Principle: What is done on one side of the equation must be done on the other side to maintain balance.
Example: Solve (x + 3 = 7)
Subtract 3 from both sides: (x + 3 - 3 = 7 - 3)
Simplified: (x = 4)
Transposing Method
Principle: Move terms from one side of the equation to the other by changing their signs.
Example: Solve (2x – 4 = 10)
Add 4 to both sides: (2x - 4 + 4 = 10 + 4)
Simplified: (2x = 14)
Divide both sides by 2: (x = 7)
Applications of Simple Equations
Solving Word Problems
Example: “The sum of three times a number and 11 is 32. Find the number.”
Formulate the equation: (3x + 11 = 32)
Solve the equation:
Subtract 11 from both sides: (3x = 21)
Divide both sides by 3: (x = 7)
More Complex Equations
Equations with Variables on Both Sides
Example: Solve (3x + 5 = 2x + 9)
Subtract (2x) from both sides: (3x - 2x + 5 = 9)
Simplified: (x + 5 = 9)
Subtract 5 from both sides: (x = 4)
Equations Involving Fractions
Example: Solve (frac{x}{2} + 3 = 7)
Subtract 3 from both sides: (frac{x}{2} = 4)
Multiply both sides by 2: (x = 8)
Verification of Solutions
Substitution Method
Principle: Substitute the solution back into the original equation to verify.
Example: Verify (x = 4) for (3x + 5 = 17)
Substitute (x = 4): (3(4) + 5 = 12 + 5 = 17)
Verification: (17 = 17), solution is correct.
Practical Applications
Real-life Applications of Simple Equations
Example: Budgeting, calculating distances, and determining ages in word problems.
Conclusion
Summary of the Chapter
The chapter “Simple Equations” provides a comprehensive understanding of formulating, solving, and verifying algebraic equations. It covers various methods and applications, emphasizing the importance of equations in solving real-world problems.
Additional Information
Key Terms and Concepts
Equation: A statement that shows the equality of two expressions.
Variable: A symbol (often (x)) that represents an unknown value.
Transposing: Moving terms from one side of the equation to the other by changing their signs.
Balancing: Ensuring both sides of the equation remain equal by performing the same operations on both sides.
Important Examples and Cases
Formulating Equations: Translating word problems into algebraic equations.
Solving Equations: Using balancing and transposing methods.
Verification: Substituting solutions back into the original equation to verify correctness.
Notable Observations
Importance of Simple Equations: Fundamental for understanding more complex algebraic concepts.
Applications in Various Fields: Widely used in budgeting, science, engineering, and everyday problem-solving.
FAQs
1. What is an equation?
An equation is a mathematical statement that shows the equality of two expressions separated by an equals sign (=).
2. Give an example of a simple equation.
An example of a simple equation is ( 2x + 3 = 7 ).
3. How do you solve the equation ( x + 5 = 12 )?
Subtract 5 from both sides to get ( x = 7 ).
4. What is the method of balancing in solving equations?
The method of balancing involves performing the same operation on both sides of the equation to maintain equality.
5. Solve the equation ( 3x – 4 = 8 ) using the transposing method.
Add 4 to both sides: ( 3x = 12 ), then divide by 3: ( x = 4 ).
6. How do you verify the solution of an equation?
Substitute the solution back into the original equation to check if both sides are equal.
7. What is a variable in an equation?
A variable is a symbol (often ( x )) that represents an unknown value in an equation.
8. Translate the word problem “A number increased by 7 is 15” into an equation.
The equation is ( x + 7 = 15 ).
9. Solve the equation ( 5x + 2 = 17 ).
Subtract 2 from both sides: ( 5x = 15 ), then divide by 5: ( x = 3 ).
10. What does it mean to transpose a term in an equation?
Transposing means moving a term from one side of the equation to the other by changing its sign.
11. Solve the equation ( x/4 – 3 = 2 ).
Add 3 to both sides: ( x/4 = 5 ), then multiply by 4: ( x = 20 ).
12. How do you solve an equation with variables on both sides, such as ( 3x + 5 = 2x + 9 )?
Subtract ( 2x ) from both sides: ( x + 5 = 9 ), then subtract 5: ( x = 4 ).
13. What is the solution to the equation ( 2x + 6 = 4x – 8 )?
Subtract ( 2x ) from both sides: ( 6 = 2x – 8 ), add 8 to both sides: ( 14 = 2x ), then divide by 2: ( x = 7 ).
14. Solve the equation ( 2x/3 = 6 ).
Multiply both sides by 3: ( 2x = 18 ), then divide by 2: ( x = 9 ).
15. What is the importance of simple equations in real life?
Simple equations are used in various fields such as budgeting, science, engineering, and everyday problem-solving.
16. How do you check if a given value is a solution to an equation?
Substitute the value into the equation and check if both sides are equal.
17. Solve the word problem: “Three times a number decreased by 2 is 16.”
Formulate the equation: ( 3x – 2 = 16 ), add 2 to both sides: ( 3x = 18 ), then divide by 3: ( x = 6 ).
18. What is the balancing principle in equations?
The balancing principle states that what is done on one side of the equation must be done on the other side to maintain balance.
19. Translate the word problem: “The sum of a number and 8 is 20” into an equation.
The equation is ( x + 8 = 20 ).
20. Solve the equation ( 4x – 7 = 9 ).
Add 7 to both sides: ( 4x = 16 ), then divide by 4: ( x = 4 ).
21. What is the substitution method in verifying solutions?
The substitution method involves substituting the solution back into the original equation to verify correctness.
22. Solve the equation ( 7 – 2x = 1 ).
Subtract 7 from both sides: ( -2x = -6 ), then divide by -2: ( x = 3 ).
23. What does it mean if an equation has no solution?
An equation has no solution if no value of the variable can satisfy the equation.
24. Solve the equation ( x/5 + 3 = 6 ).
Subtract 3 from both sides: ( x/5 = 3 ), then multiply by 5: ( x = 15 ).
25. What is the equation for “Five times a number is equal to twenty-five”?
The equation is ( 5x = 25 ).
Chapter 4:Simple Equations MCQs
1. What is an equation?
2. What is the solution to the equation (x + 7 = 15)?
3. What is the median of the data set [15, 20, 35, 40, 50]?
4. What is the method of balancing in solving equations?
5. Solve the equation (3x – 4 = 8).
6. How do you verify the solution (x = 4) for the equation (3x + 5 = 17)?
7. What is the variable in the equation (2x + 3 = 9)?
8. Translate the word problem “A number increased by 7 is 15” into an equation.
9. Solve the equation (5x + 2 = 17).
10. What does it mean to transpose a term in an equation?
11. Solve the equation (x/4 – 3 = 2).
12. How do you solve an equation with variables on both sides, such as (3x + 5 = 2x + 9)?
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