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 ↓


Graph of Linear Equations


In the study of algebra, particularly focusing on linear equations in two variables, the concept of graphing these equations is foundational. By plotting linear equations on a coordinate plane, we visualize the relationship between two variables, usually represented as x and y. This graphical representation helps to understand the nature of linear equations and provides important insights into their behavior and solutions. In this detailed guide, we will explore the graph of linear equations, its key features, and how it can be used effectively to understand linear relationships.

Introduction to linear equations in two variables

A linear equation in two variables is an equation that forms a straight line when graphed on the coordinate plane. The general form of a linear equation in two variables is:

axis + by = c

In this equation, A, B, and C are constants, while x and y are variables. A typical example of a linear equation is:

2x + 3y = 6

The objective is to find all the points (x, y) that satisfy this equation. These points, when plotted on the graph, will fall on a straight line.

Understanding the coordinate plane

The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal line (the x-axis) and a vertical line (the y-axis). The point where these axes intersect is known as the origin, represented by (0, 0). Each point on the coordinate plane is identified by an ordered pair (x, y), where x represents the horizontal distance from the origin and y represents the vertical distance.

Graphing a linear equation

To graph a linear equation, you need to plot the points that satisfy the equation and then connect these points with a straight line. Here is how you can graph the equation 2x + 3y = 6 step by step:

Step 1: Find the intercepts

Finding intercepts is a common method for graphing linear equations.

X-intercept

The x-intercept is the point at which the graph crosses the x-axis. At this point, the value of y is 0. Substitute y = 0 into the equation:

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

Thus, the x-intercept is (3, 0).

Y-intercept

The y-intercept is where the graph crosses the y-axis. At this point, the value of x is 0. Substitute x = 0 into the equation:

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

Thus, the y-intercept is (0, 2).

Step 2: Plot the intercepts

Plot the x-intercept (3, 0) and the y-intercept (0, 2) on the coordinate plane. These points are fundamental because only two points are needed to plot a straight line. Connect these points to graph the equation.

(3, 0) (0, 2)

Step 3: Verify the line

To ensure accuracy, choose another point on the line by substituting a different value for x. If the point satisfies the equation, this confirms the correctness of the graph.

Choose x = 1:

2(1) + 3y = 6
2 + 3y = 6
3y = 4
y = 4/3

The point (1, 4/3) should lie on the line. If plotted, it will be in line with the existing points.

Equation of a line: slope-intercept form

The slope-intercept form is another way to express a linear equation, which makes graphing simpler. The formula is:

y = mx + b

In this form, m represents the slope of the line, and b is the y-intercept. Slope m represents the slope and direction of the line.

Rearrange the given equation 2x + 3y = 6 into slope-intercept form:

3y = -2x + 6
y = -(2/3)x + 2

In this form, the slope m is -2/3, and the y-intercept b is 2. This indicates that the line decreases with a slope of -2/3 and cuts the y-axis at 2.

Understanding slope

The slope of the line, which is determined by the coefficient m in slope-intercept form, is the ratio of the change in y to the change in x between two points on the line. The formula for calculating the slope between two points (x1, y1) and (x2, y2) is:

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

Consider the two points (3, 0) and (0, 2) from the first example.

m = (2 - 0) / (0 - 3)
m = 2/-3
m = -2/3

The slope -2/3 confirms that the line decreases as you move from the left to the right of the graph.

Parallel and perpendicular lines

When dealing with linear equations, it is important to recognize the relationship between the lines. Two lines can be either parallel or perpendicular:

Parallel lines

Two lines are parallel if they have the same slope. For example, the lines y = 2x + 3 and y = 2x - 4 are parallel because they both have a slope of 2.

Perpendicular lines

Two lines are perpendicular if the product of their slopes is -1. For example, the lines y = (1/2)x + 3 and y = -2x + 1 are perpendicular because:

(1/2) * -2 = -1

Creating a graph using slope and y-intercept

Let's verify and graph the line using the slope-intercept form:

For the equation y = -(2/3)x + 2:

1. Start by drawing the y-intercept (0, 2) on the graph.

2. Use the slope -2/3 (rise over run): From the y-intercept, move 2 units down (rise) and 3 units to the right (run).

3. Mark this new point and draw a line through both points.

(0, 2) (3, 0)

Applications of graphing of linear equations

Graphing linear equations is not only an academic exercise but also has practical applications in a variety of fields:

  • Economics: Understanding supply and demand curves, cost functions.
  • Physics: Describing motion using velocity vs. time graphs.
  • Engineering: Drawing of stress-strain relationships, electrical circuit analysis.
  • Statistics: Conducting linear regression analysis for predictions and trends.

Practice problems

1. Graph the equation 4x - y = 8 and identify the intercepts.

Solution: Convert to slope-intercept form y = 4x - 8, find intercept, plot point.

2. Determine whether the lines y = 3x + 5 and y = -1/3x - 2 are perpendicular.

Solution: Calculate the product of the slopes 3 * -1/3 = -1, verify perpendicularity.

3. Write the equation of the line with slope 5 passing through the point (2, 1).

Solution: Use the point-slope form y - y1 = m(x - x1): y - 1 = 5(x - 2).

Conclusion

Graphing linear equations is a powerful tool for understanding linear relationships between variables. By mastering this topic, you unlock the ability to visualize equations, solve systems graphically, and apply these principles to real-world scenarios. Keep practicing by plotting different linear equations and exploring their graphs to deepen your understanding and intuition about linear relationships.


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Graph of Linear Equations

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