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 ↓


Application of Linear Equations in Word Problems


Linear equations in two variables are an important part of algebra that you learn in Class 10 Maths. These equations are important because they apply to real-world situations and help solve everyday problems. Understanding these equations can greatly enhance your problem-solving skills and logical thinking. This article will provide a comprehensive overview of how linear equations in two variables can be applied to word problems using simple language and many examples to aid understanding.

Understanding linear equations in two variables

A linear equation in two variables is generally of the form:

ax + by = c

Here, 'a', 'b' and 'c' are constants, while 'x' and 'y' are variables. This form of the equation when plotted graphically is a straight line.

The main goal in word problems is to form such equations using the given data and conditions, and then find the values of 'x' and 'y' by solving them.

How to develop equations from word problems

Understanding how to convert a problem into equations is an important step in solving word problems. Let's consider a general approach to converting word problems into linear equations:

  1. Read the problem carefully: Understand what is being asked. Identify the quantities involved and what you need to find out.
  2. Define variables: Assign symbols like 'x' and 'y' to the unknown values you need to find.
  3. Convert words into equations: Use the information provided to create an equation. Pay attention to key phrases that can help you identify mathematical operations.
  4. Solve equations: Use algebraic methods such as substitution or elimination to find the value of unknown values.
  5. Check your solution: Verify the solution by re-substituting the original conditions into the word problem.

Common scenarios in word problems

Solving linear equations using word problems can involve a variety of scenarios. Here are some common scenarios and how linear equations can be applied:

1. Age-related problems

Age-related problems usually involve relationships between the ages of different people. For example:

Example problem: Sally is two years older than twice the age of her brother John. If the sum of their ages is 22, how old is each of them?

Solution:

  1. Define the variables: Let 'x' be John's age and 'y' be Sally's age.
  2. Develop the equations: From the problem statement, we have the equations:
    y = 2x + 2
    x + y = 22
  3. Solve the equation:
    Substitute y = 2x + 2 from the first equation into the second equation:
    x + (2x + 2) = 22
    3x + 2 = 22
    3x = 20
    x = 20/3
  4. Find y: Use the value of 'x' to find 'y':
    y = 2(20/3) + 2
    y = 40/3 + 6/3
    y = 46/3

Both excerpts show their correct ages. Be sure to verify the reasonableness of the answers based on your interpretation of the problem.

2. Mixing problems

These problems often involve finding the proportions of different substances in a mixture. Consider this example:

Example problem: A chemist needs to mix a 10% acid solution with a 50% acid solution to obtain 200 milliliters of a 30% acid solution. How much of each solution should he use?

Solution:

  1. Define the variables: Let's say 'x' is the volume of the 10% solution and 'y' is the volume of the 50% solution.
  2. Develop the equation:
    x + y = 200
    0.1x + 0.5y = 0.3(200)
  3. Solve the equation:
    4. First equation gives: y = 200 - x
    Substitute into the second equation:
    0.1x + 0.5(200 - x) = 60
    0.1x + 100 - 0.5x = 60
    -0.4x = -40
    x = 100
  4. Find y: x = 100 Re-substitute:
    y = 200 - 100
    y = 100

The chemist should use 100 ml of each solution.

Graphical representation

Visualization can be of great help in understanding the solution of linear equations. Let us present a graphical example:

ax + by = c

This image shows the intersection of these lines that represent x = 3 and y = 2 when plotted as ax + by = c. The intersection represents the solution to both linear equations.

Conclusion

Linear equations in two variables have many applications when it comes to solving word problems. Understanding the formation of equations based on problem statements, translating them into visual or graphical representations, and solving them to find the required solutions are key components of the grade 10 math curriculum. Getting familiar with exercises and varied examples is helpful in mastering these skills.

The problems and examples discussed here serve as a basic foundation for understanding the practical utility of algebraic concepts in real-world scenarios. By mastering these techniques, you can move from solving abstract equations to understanding concrete situations.


Grade 10 → 2.2.3


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Application of Linear Equations in Word Problems

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