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  • Let’s explore another problem that is commonly asked in the CBSE Class 9 exam pattern…

    Question

    The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are reversed. Find the number.

    Solution

    Step 1: Define the variables

    Let the two-digit number be 10x+y10x + y, where:

    • xx is the digit in the tens place.
    • yy is the digit in the units place.

    Step 2: Write the given conditions as equations

    1. The sum of the digits is 9: x+y=9.(1)x + y = 9 \tag{1}.
    2. If 27 is subtracted from the number, the digits are reversed: 10x+y−27=10y+x⇒9x−9y=27⇒x−y=3.(2)10x + y – 27 = 10y + x \quad \Rightarrow \quad 9x – 9y = 27 \quad \Rightarrow \quad x – y = 3 \tag{2}.

    Step 3: Solve the equations

    From equation (2): x−y=3⇒x=y+3.(3)x – y = 3 \quad \Rightarrow \quad x = y + 3 \tag{3}.

    Substitute x=y+3x = y + 3 into equation (1): (y+3)+y=9.(y + 3) + y = 9. 2y+3=9⇒2y=6⇒y=3.2y + 3 = 9 \quad \Rightarrow \quad 2y = 6 \quad \Rightarrow \quad y = 3.

    Substitute y=3y = 3 into equation (3): x=y+3=3+3=6.x = y + 3 = 3 + 3 = 6.

    Step 4: Write the solution

    The number is: 10x+y=10(6)+3=63.10x + y = 10(6) + 3 = 63.

    Verification

    1. The sum of the digits is: 6+3=96 + 3 = 9 (correct).
    2. Subtracting 27 from 63 gives 63−27=3663 – 27 = 36, which is the reverse of 63 (correct).

    This problem ensures the following are covered:

    1. Algebraic representation of word problems.
    2. Logical derivation and substitution.
    3. Verification of the result.

    Would you like me to graphically represent another scenario or explain additional examples? PLZ

    COOMENT…..

  • “Linear Equations in Two Variables,” is pivotal for understanding mathematical relationships and their graphical representations..

    Strengths of the Chapter

    1. Foundation for Advanced Topics:
      • The chapter introduces the concept of linear equations, forming a base for higher mathematics topics like coordinate geometry, calculus, and linear programming.
    2. Practical Applications:
      • Linear equations have numerous real-life applications in fields like economics, engineering, physics, and business analytics.
      • Examples include calculating taxi fares, budgeting, or understanding relationships in data.
    3. Graphical Understanding:
      • By plotting equations on graphs, students develop spatial reasoning and a visual understanding of algebraic relationships.
    4. Interactive Learning:
      • Solving equations graphically and algebraically makes the topic engaging and enhances problem-solving skills.

    Challenges and Criticisms

    1. Abstract Nature:
      • For some students, understanding the relationship between algebraic equations and their graphical representation can be challenging.
    2. Insufficient Real-Life Context:
      • While the chapter mentions applications, it does not delve deeply into practical, relatable scenarios, making it seem disconnected from real-world use.
    3. Graphical Limitations:
      • The textbook heavily relies on manual plotting, which can be tedious. Incorporating technology like graphing calculators or software could modernize the learning experience.
    4. Simplistic Problems:
      • Many problems focus on straightforward equations, lacking complexity or real-world data that could engage more advanced learners.

    Suggestions for Improvement

    1. Integrate Technology:
      • Encourage the use of graphing tools like GeoGebra or Desmos to make learning dynamic and interactive. This also prepares students for modern mathematical practices.
    2. Real-World Problems:
      • Incorporate more practical problems, such as interpreting graphs in weather forecasting, financial modeling, or urban planning.
    3. Introduce Non-Linear Systems:
      • Briefly touch upon non-linear systems to contrast and deepen understanding of linear equations.
    4. Collaborative Projects:
      • Include projects like surveying distances and costs or analyzing data trends to show how linear equations are used in data analysis.

    Higher-Order Questions

    1. Critical Thinking:
      • How would the graph of 2x+3y−6=02x + 3y – 6 = 0 change if the coefficient of xx were doubled?
    2. Real-Life Application:
      • A company’s revenue and expenses are modeled by R(x)=50xR(x) = 50x and E(x)=30x+200E(x) = 30x + 200, where xx is the number of units sold. Find the break-even point.
    3. Interdisciplinary Connection:
      • How can linear equations be used in science experiments to predict outcomes based on variable changes?

    Conclusion

    The chapter serves as an essential building block for mathematical reasoning and problem-solving. However, introducing modern tools, real-world applications, and interdisciplinary approaches would make it more engaging and relevant for students. This would not only solidify their understanding but also foster an appreciation for the practical significance of linear equations.

    Let me know if you’d like further insights or practice problems!

    COOMMENT REGARDING YOUR SPECIFIC QUESTION ….

  • The Fourth Chapter of CBSE Class 9 Maths “Linear Equations in Two Variables”

    Concepts

    1. Linear Equation in Two Variables:
      A linear equation in two variables is an equation of the form: ax+by+c=0ax + by + c = 0 where:
      • a,b,a, b, and cc are real numbers.
      • xx and yy are variables.
      Example: 2x+3y−5=02x + 3y – 5 = 0.
    1. Solution of a Linear Equation in Two Variables:
      A solution is a pair of values (x,y)(x, y) that satisfy the equation. For example, for 2x+3y−5=02x + 3y – 5 = 0:
      Substituting x=1,y=1x = 1, y = 1: 2(1)+3(1)−5=0(True).2(1) + 3(1) – 5 = 0 \quad \text{(True)}.
    1. Graphical Representation:
      • A linear equation in two variables represents a straight line on a graph.
      • To draw the graph:
        1. Rewrite the equation in the form y=mx+cy = mx + c (slope-intercept form).
        2. Find at least two solutions.
        3. Plot the solutions on the graph.
        4. Join the points to form a straight line.

    Solved Examples

    Example 1: Plotting the Graph

    Problem: Plot the graph of 3x+2y=63x + 2y = 6.

    Solution:
    Rewrite as y=−32x+3y = \frac{-3}{2}x + 3.
    Find solutions by substituting values of xx:

    xxyy
    0062=3\frac{6}{2} = 3
    22−3(2)2+3=0\frac{-3(2)}{2} + 3 = 0
    44−3(4)2+3=−3\frac{-3(4)}{2} + 3 = -3

    Plot points (0,3)(0, 3), (2,0)(2, 0), and (4,−3)(4, -3), then join them to form a line.

    Example 2: Checking Solutions

    Problem: Check whether (1,2)(1, 2) is a solution of 2x−y+3=02x – y + 3 = 0.

    Solution: Substitute x=1,y=2x = 1, y = 2: 2(1)−2+3=0(False).2(1) – 2 + 3 = 0 \quad \text{(False)}.

    Answer: (1,2)(1, 2) is not a solution.

    Example 3: Word Problem

    Problem: A taxi company charges a fixed rate of ₹50 plus ₹15 per kilometer. Write a linear equation and find the cost of traveling 10 km.

    Solution:

    • Let yy be the cost and xx be the distance (in km).
    • Equation: y=15x+50.y = 15x + 50.
    • For x=10x = 10: y=15(10)+50=150+50=200.y = 15(10) + 50 = 150 + 50 = 200.

    Answer: Cost = ₹200.

    Applications

    1. Economics:
      Linear equations model relationships between costs and revenues, such as profit/loss analysis.
    2. Physics:
      Used to represent uniform motion or constant acceleration scenarios.
    3. Business and Finance:
      Helps determine supply-demand relationships, fixed costs, and variable costs.
    4. Computer Programming:
      Algorithms use linear equations for graphical representations and optimizations.

    COMMENTS ON YOUR QUESTION REGARDING LINEAR EQUATIONS..