Linear Equations Solving Methods in Class 9 Mathematics
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Questions and Answers

What is the solution to the equation $3-\frac{x}{5}=7$?

  • $x=15$
  • $x=5$ (correct)
  • $x=10$
  • $x=20$
  • In the example given, what does the ratio $\frac{40, \text{km}}{30, \text{kmh}^{-1}}$ represent?

  • Speed of 30 km per hour
  • Time taken to travel 40 km (correct)
  • Distance traveled in 40 minutes
  • Time taken to travel 30 km
  • In the equation $x=20-15$, what does $x$ represent?

  • The time taken to travel 40 km
  • The speed of travel
  • The solution to the equation (correct)
  • The number of apples sold
  • In the context of linear equations, what do higher-level concepts like quadratic equations and systems of linear equations involve?

    <p>Multiple variables and relationships</p> Signup and view all the answers

    Why are linear equations important in understanding topics like quadratic equations and graphical representation of data?

    <p>To build a foundation for understanding more complex concepts</p> Signup and view all the answers

    What is the first step in the two-step method for solving linear equations?

    <p>Simplify each term without changing its sign</p> Signup and view all the answers

    Which method is effective for solving one variable equations with only whole numbers or integers involved?

    <p>One Step Method</p> Signup and view all the answers

    In the equation 7x - 6 = 8, what should be done after dividing both sides by 7 following the one-step method?

    <p>Add 6 to both sides</p> Signup and view all the answers

    What should be moved to one side while placing constants to another side in the one-step method of solving linear equations?

    <p>Terms containing x</p> Signup and view all the answers

    How should the equation be set up in the two-step method so that the coefficient of x equals zero?

    <p>Divide the constant part by the coefficient of x</p> Signup and view all the answers

    Study Notes

    Solving Linear Equations in Class 9th PSEB Mathematics

    In your mathematics journey through Class 9 with Punjab School Education Board (PSEB), you'll encounter linear equations — foundational building blocks of algebra and problem-solving skills. In this section, let's explore how to solve these simple yet powerful expressions that describe real life situations using two fundamental methods.

    One Step Method

    This method is effective when solving one variable equations like (ax + b = c) where coefficients ((a,\ b,\ c)) involve only whole numbers or integers. Here's how it works:

    • If there's a coefficient before the x term, isolate it by dividing both sides by the number, i.e., set [ax = c - b]
    • Next, move all terms containing x to one side while placing constants to another side: [x = \frac{c - b}{a}]

    Example: Solve (7x - 6= 8.) First, divide both sides by 7: [ x=\frac{8+6}{7} ] Then simplify: [ x=\frac{14}{7}=\frac{2\times 7}{7}=2 ]

    Two Step Method

    The two step method allows us to solve more complex linear equations involving fractions, decimals, or multiplication and division operations within parentheses. It involves following these steps:

    1. Simplify each term without changing its sign.
    2. Perform any operation inside parentheses first.
    3. Move all terms containing (x) to one side while keeping their signs intact.
    4. Set up the equation so that the coefficient of (x) equals zero. Then, find the constant part divided by the coefficient of (x).

    Let's apply our newfound knowledge to some examples:

    Solve (3-\dfrac{x}{5}=7.)

    Firstly, reverse the order of the fraction term and multiply the entire equation by 5: [ 5\left( 3-\frac{x}{5} \right)=5(7)\quad\Rightarrow\quad 15-x,= ,35 ] Next, add (x) to both sides: [ x=20-15 \quad\Rightarrow \quad x=5 ]

    Applications and Real Life Situations

    Linear relationships often represent proportions, rates, slopes of lines, and distance-time graphs. For instance:

    • A shopkeeper sells apples at Rs.25 per dozen ((12)); if he earns Rs.300 selling them, determine how many dozens were sold. To solve this, we can create a proportion: (\frac{\text{number of apples}}{12}=\frac{Rs300}{25}.)

    • Determine the time taken to travel 40 km at a speed of 30kmph. To do this, formulate a ratio: [\frac{\text{distance}}{\text{speed}}=\left(\frac{40,\text{km}}{30,\text{kmh}^{-1}}\right)=\left(\frac{4}{3},\text{hour}\right)^{-1}=\left(\frac{3}{4},\text{hour}\right),] which indicates 0.75 hours or (45\ minutes.)

    As you progress further into Class 9 math, keep practicing problems related to linear equations; they will help you understand higher level concepts such as quadratic equations, systems of linear equations, and graphical representation of data much easier!

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    Description

    Explore the one-step and two-step methods for solving linear equations in 9th Grade Mathematics with Punjab School Education Board (PSEB). Learn how to manipulate equations involving whole numbers, fractions, and decimals to find solutions. Apply these techniques to real-life scenarios involving proportions, rates, and distance-time graphs.

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