Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Linear Equations in Two Variables ↓


Algebraic Method


The algebraic method is one of the fundamental techniques used to solve linear equations in two variables. Understanding this method is important for solving problems where two different conditions need to be satisfied simultaneously, often represented as two linear equations. In this document, we will explore the concept, step-by-step procedures, examples, and various aspects of the algebraic method.

Introduction to linear equations in two variables

A linear equation in two variables is an equation that can be written in the following form:

ax + by = c

where x and y are variables, and a, b and c are constants. The solution to these equations is a set of values of x and y that makes the equation true.

What is the algebraic method?

The algebraic method involves using algebra techniques to eliminate one variable, allowing you to solve for the other. There are mainly two approaches within the algebraic method:

  1. Replacement method
  2. Elimination method

1. Substitution method

The substitution method involves solving one of the equations for a single variable and substituting this expression into the other equation. This turns the system of equations into a single equation in one variable. Let's look at the steps:

  1. Solve an equation for one of the following variables:

    Pick any equation and solve for one variable in terms of the other. For example, if you have this equation:

    x + 2y = 5

    You can solve for x like this:

    x = 5 - 2y
  2. Substitute this expression into the other equation:

    Substitute the expression for the variable found in the first step into the second equation. Continuing our example, if the second equation is:

    3x - y = 4

    Substitute x = 5 - 2y in the equation:

    3(5 - 2y) - y = 4
  3. Solve the resulting single-variable equation:

    Simplify and solve for the remaining variable:

    15 - 6y - y = 4
    15 - 7y = 4
    -7y = 4 - 15
    -7y = -11
    y = 11/7
  4. Substitute again to find the other variables:

    Substitute the value of y into the expression obtained in Step 1 to find x :

    x = 5 - 2(11/7)

    Simplifying this, you get:

    x = 5 - 22/7
    x = 35/7 - 22/7
    x = 13/7

    Thus, the solution is (13/7, 11/7).

0 X Y

2. Elimination method

The elimination method involves aligning two linear equations in such a way that adding or subtracting them eliminates one of the variables. This allows you to solve the remaining single-variable equation. The basic steps for this method include:

  1. Align the equations:

    Write both equations in standard form and try to align them to focus on eliminating one variable. Let's consider:

    2x + 3y = 8
    4x - 9y = -2
  2. Equalize the coefficients of one variable:

    Multiply one or both equations by constants to make the coefficients of a variable equal (or opposite). For example, multiply the first equation by 2:

    4x + 6y = 16

    Now, subtract the second equation from this:

    (4x + 6y) - (4x - 9y) = (16) - (-2)
  3. Eliminate one variable and solve for the other:

    Simplify the result:

    4x + 6y - 4x + 9y = 16 + 2
    15y = 18
    y = 18/15
    y = 6/5
  4. Substitute again to find the other variables:

    Substitute the value of y into one of the original equations. Let's choose the first equation:

    2x + 3(6/5) = 8

    On simplifying, we get:

    2x + 18/5 = 8
    2x = 8 - 18/5
    2x = 40/5 - 18/5
    2x = 22/5
    x = 11/5

    Hence the solution is (11/5, 6/5).

0 X Y

Conclusion

Understanding the algebraic methods of substitution and elimination is essential to solving systems of linear equations. Both methods rely on simplifying the problem into a single equation with one variable, making it easier to solve. Practicing with these methods enhances problem-solving skills and lays a strong foundation for more advanced algebra topics.

As you work on these problems, you will discover the beauty of algebra and its ability to describe and solve relationships between variables using clear, logical processes.


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