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 → Coordinate Geometry → Equation of a Line ↓


Slope-Intercept Form


In the world of coordinate geometry, one of the most common ways to represent a line is using the slope-intercept form. This form is a simple but powerful tool that helps us understand and work with linear equations. The beauty of this form is in its simplicity, which makes it easy to understand and apply to a variety of problems. The general expression for the slope-intercept form of a line is:

y = mx + c

In this equation:

  • y is the dependent variable, usually represented by a vertical position on the graph.
  • x is the independent variable, usually represented by a horizontal position on the graph.
  • m is the slope of the line.
  • c is the y-intercept of the line.

Understanding each component

1. Slope (m)

The slope of a line is a number that shows both the direction and the slope of the line. In mathematics, the slope is usually represented by the letter m. The slope can be calculated by dividing the change in y by the change in x between any two points on the line. This is often referred to as "rise over run."

m = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)

Here, x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point. Let's visualize this:

(x1, y1) (x2, y2)

Here, the slope shows how the line goes up and to the right. If the slope is positive, the line goes up as you move from left to right. If the slope is negative, the line goes down.

2. Y-intercept (c)

The y-intercept of a line is the point where the line crosses the y-axis. The value of c gives this particular point, which occurs when x equals zero. Thus, the equation of the line when it crosses the y-axis becomes:

y = c

Let's visualize the y-intercept:

(0, C)

At the marked point, the line crosses the y-axis. This is our y-intercept c.

Discovery of the equation

Consider the equation:

y = 2x + 3

Here, the slope m is 2, which means that for every unit increase in x, y increases by 2 units. The y-intercept c is 3, which means that the line cuts the y-axis at the point (0, 3).

Consider another example:

y = -4x + 1

In this case, the slope is m -4, which indicates that for every unit increase in x, y decreases by 4 units. The y-intercept c is 1.

Working with slope-intercept form

Converting points into equations

If we know two points through which a line passes, we can find its slope and then write the equation in slope-intercept form.

Suppose we have points (1, 2) and (3, 6). First calculate the slope m:

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

Now use the point-slope form to find the line:

y - y1 = m(x - x1)

Taking the point (1, 2):

y - 2 = 2(x - 1)

Simplify it:

y = 2x

Now, use the second point for verification. Inserting (3, 6) in the equation confirms the solution.

Application

The slope-intercept form is primarily used to graph a line. Using the slope and y-intercept, one can quickly create a graph. It is especially useful in real-world scenarios, such as:

  • Predicting trends in data sets.
  • Solving problems involving linear relationships in physics and engineering.

Graphing a line

To graph the line with the equation y = mx + c, follow these steps:

  1. Start at the y-intercept point (0, c).
  2. Find the second point using the slope m. If m is a fraction, rise / run can guide you. From the y-intercept, move vertically (rise) and horizontally (run) to find the next point.
  3. Draw a line through the points obtained.

Example graph:

y = 2x + 1
(0, 1)

We start at (0, 1) and follow the slope 2 to reach the next point 2 up and 1 across.

Conclusion

The slope-intercept form y = mx + c is an essential concept in mathematics, providing a fundamental understanding of linear equations and graphing. The simplicity of this form allows for easy interpretation and application in a variety of fields. Whether used to solve academic problems or to model real-world situations, this form serves as a foundational tool in both educational and practical environments.


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Slope-Intercept Form

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