Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra ↓


Linear Equations in Two Variables


Linear equations in two variables are a fundamental concept in mathematics. They are equations that involve two different variables, usually denoted as x and y. The general form of a linear equation in two variables is:

ax + by = c

In this equation, a, b and c are constants, and x and y are variables. The coefficients a and b are numbers multiplied by x and y, respectively, and c is a constant term.

Let us analyse this:

  • Linear: This means that the equation represents a straight line when graphed on the coordinate plane.
  • Equation: An equation is a mathematical statement that asserts the equality of two expressions.
  • Two variables: The equation involves two unknowns or variables.

In a linear equation in two variables, every point on the line is a solution to the equation. In other words, if you have an ordered pair (x, y) that satisfies the equation, then it will lie on the line represented by the equation.

Graphical representation

The most important aspect of linear equations is their graphical representation. In two dimensions, a linear equation represents a straight line on the Cartesian plane.

X Y

In the example above, the blue line represents the graph of a linear equation. The black axis is the Cartesian coordinate system where the horizontal line is the x-axis and the vertical line is the y-axis. Any point (x, y) on the blue line is a solution to the linear equation it represents.

Finding a solution

To find a solution to a linear equation in two variables, you choose a value for one variable and solve for the other. For example, given the equation:

2x + 3y = 6

If you choose x = 0, you can solve for y:

2(0) + 3y = 6 3y = 6 y = 2

Therefore, one solution is (0, 2). Similarly, you can choose y = 0 and solve for x:

2x + 3(0) = 6 2x = 6 x = 3

The second solution is (3, 0). You can find many pairs of points that satisfy the equation, and these points will form a line representing the equation on the graph.

Obstructions

The x-intercept is the point where the graph crosses the x-axis. At this point, the value of y is zero. The y-intercept is the point where the graph crosses the y-axis, where the value of x is zero.

For the equation ax + by = c, the intercepts can be found as follows:

  • To find the y-intercept, set x = 0 and solve for y.
  • Substitute y = 0 and solve for x to find the x-intercept.

For example, consider the equation 4x + 5y = 20:

Finding the Y-Intercept:

4(0) + 5y = 20 5y = 20 y = 4

The y-intercept is (0, 4).

Finding the x-intercept:

4x + 5(0) = 20 4x = 20 x = 5

The x-intercept is (5, 0).

Slope of the line

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y (vertical change) to the change in x (horizontal change) between any two points on the line. Mathematically, the slope m is given as:

m = (y2 - y1) / (x2 - x1)

For example, find the slope of the line passing through the points (1, 2) and (3, 6):

m = (6 - 2) / (3 - 1) = 4 / 2 = 2

Therefore, the slope of the line is 2. A positive slope indicates that the line is rising as we move from left to right, while a negative slope indicates that the line is falling.

Standard form and slope-intercept form

Linear equations can be expressed in different forms. Two common forms are standard form and slope-intercept form.

  • Standard Form: Ax + By = C, where A, B, and C are integers, and A must be nonnegative.
  • Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

You can convert between these forms using algebraic manipulation. For example, to convert a standard form equation into slope-intercept form, solve for y:

4x + 3y = 12 3y = -4x + 12 y = -4/3x + 4

Solving systems of linear equations

Sometimes, you need to solve a system of linear equations, which consists of two or more linear equations. There are several ways to solve systems of equations, such as:

  • Graphical Method: Graph each equation and find the intersection points.
  • Substitution method: Solve one equation for one variable, then substitute it into the other equation.
  • Elimination method: Add or subtract equations to eliminate one variable, then solve for the other.

Consider a system of equations:

2x + y = 5 x - y = 1

Uses of the substitution method:

From the second equation: x = y + 1 Substitute into the first equation: 2(y + 1) + y = 5 2y + 2 + y = 5 3y = 3 y = 1 x = 2

The solution of the system is (2, 1).

Applications of linear equations

Linear equations in two variables appear in a variety of real-world situations. Some applications include:

  • Business: Finding the break-even point by calculating costs and revenue.
  • Physics: Analysis of steady motion or uniform motion.
  • Economics: Predicting demand and supply using linear models.

Understanding linear equations helps in solving practical problems where the relationship between two quantities is linear or directly proportional.

Conclusion

Linear equations in two variables form the basis for understanding more complex algebraic and geometric concepts. Understanding how these equations represent lines on a graph, finding solutions, and applying them to real-life situations is important for students. By mastering this topic, learners can confidently move on to more sophisticated mathematical explorations.


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Linear Equations in Two Variables

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