Linear Equations Application Practice Question

Duration: 33 min

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AI Summary

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This educational video presents a comprehensive tutorial on solving real-world problems using linear equations. The instructor systematically works through ten distinct word problems, demonstrating a consistent problem-solving methodology. The core approach involves identifying unknowns, translating the problem's conditions into algebraic equations, and solving them using standard techniques. The problems cover a range of applications, including finding two numbers based on their sum and difference, relationships between numbers (e.g., one being twice another), consecutive odd numbers, and problems involving the product of numbers. The video also includes a problem on the sum of a number and its reciprocal, which leads to a quadratic equation, and a classic distance-speed-time problem. The final problem involves a fraction where the denominator is related to the numerator. The entire lesson is delivered on a digital whiteboard, with the instructor writing out each step of the solution process, making it a clear and practical guide for students learning to apply linear equations.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title slide reading "Linear Equation Application". The instructor then introduces the first problem, Q1, which states: "The sum of two numbers is 45, and their difference is 9. Find the numbers." The instructor begins the solution by defining the two numbers as 'x' and 'y'. He sets up the first equation as x + y = 45 and the second as x - y = 9. He then adds the two equations together to eliminate 'y', resulting in 2x = 54. Solving for x gives x = 27. He then substitutes x = 27 into the first equation to find y = 18. The final answer is presented as the two numbers being 27 and 18.

  2. 2:00 – 5:00 02:00-05:00

    The instructor moves to the second problem, Q2: "The larger of two numbers is twice the smaller. Their sum is 24. Find the numbers." He defines the smaller number as 'n' and the larger as '2n'. He sets up the equation n + 2n = 24, which simplifies to 3n = 24. Solving for n gives n = 8. Therefore, the smaller number is 8 and the larger is 16. The third problem, Q3, is presented: "A number is 5 more than half of another number. Their total is 23. Find both numbers." He defines the smaller number as 'n' and the larger as 'n/2 + 5'. The equation becomes n + (n/2 + 5) = 23. He combines like terms to get (3n/2) + 5 = 23, then subtracts 5 to get 3n/2 = 18. Multiplying both sides by 2 gives 3n = 36, and solving for n gives n = 12. The smaller number is 12, and the larger is 11.

  3. 5:00 – 10:00 05:00-10:00

    The fourth problem, Q4, is introduced: "The sum of two consecutive odd numbers is 36. Find the numbers." The instructor explains that consecutive odd numbers differ by 2, so he defines the first number as 'n' and the second as 'n+2'. The equation is n + (n+2) = 36, which simplifies to 2n + 2 = 36. Subtracting 2 gives 2n = 34, and solving for n gives n = 17. The numbers are 17 and 19. The fifth problem, Q5, states: "One number is 3 more than the other. Their product is 70. Find the numbers." He defines the smaller number as 'n' and the larger as 'n+3'. The equation is n(n+3) = 70, which expands to n² + 3n = 70. He rearranges it to the standard quadratic form: n² + 3n - 70 = 0. He factors this as (n+10)(n-7) = 0, yielding solutions n = -10 or n = 7. The positive solution is n = 7, so the numbers are 7 and 10.

  4. 10:00 – 15:00 10:00-15:00

    The sixth problem, Q6, is presented: "The perimeter of a rectangle is 48 cm. Length is 4 cm more than width. Find its dimensions." The instructor draws a rectangle and defines the width as 'w' and the length as 'l = w + 4'. The perimeter formula is 2(l + w) = 48. Substituting the length gives 2((w+4) + w) = 48, which simplifies to 2(2w + 4) = 48. This becomes 4w + 8 = 48. Subtracting 8 gives 4w = 40, and solving for w gives w = 10. The length is 14. The seventh problem, Q7, is: "The sum of a number and its reciprocal is 2.5. Find the number." He defines the number as 'n' and sets up the equation n + 1/n = 2.5. Multiplying through by 'n' gives n² + 1 = 2.5n. Rearranging to standard form yields n² - 2.5n + 1 = 0. He uses the quadratic formula to solve for n, finding the solutions n = 2 and n = 0.5.

  5. 15:00 – 20:00 15:00-20:00

    The eighth problem, Q8, is: "A number added to 7 gives 25. Find the number." The instructor sets up the equation n + 7 = 25 and solves for n, getting n = 18. The ninth problem, Q9, is a classic distance-speed-time problem: "A train travels 120 km at a certain speed. If speed were 10 km/h more, it would take 1 hour less. Find the speed." He defines the original speed as 's' km/h, so the original time is 120/s. The new speed is s+10, and the new time is 120/(s+10). The equation is 120/s - 120/(s+10) = 1. He multiplies through by s(s+10) to clear the denominators, resulting in 120(s+10) - 120s = s(s+10). This simplifies to 1200 = s² + 10s. Rearranging gives s² + 10s - 1200 = 0. He factors this as (s+40)(s-30) = 0, giving s = 30 km/h (the positive solution).

  6. 20:00 – 25:00 20:00-25:00

    The tenth and final problem, Q10, is: "The denominator of a fraction is 3 more than its numerator. If the fraction equals 2/5, find it." The instructor defines the numerator as 'n' and the denominator as 'n+3'. The equation is n/(n+3) = 2/5. He cross-multiplies to get 5n = 2(n+3), which simplifies to 5n = 2n + 6. Subtracting 2n gives 3n = 6, and solving for n gives n = 2. The numerator is 2, and the denominator is 5, so the fraction is 2/5. The video concludes with a brief summary of the topics covered, including 'Advanced Roots' and 'Foundation'.

  7. 25:00 – 30:00 25:00-30:00

    The video continues with a brief, unstructured segment where the instructor writes 'Advanced Roots' and 'Foundation' on the whiteboard. He appears to be summarizing the topics covered, possibly for a different lesson or as a concluding remark. The on-screen text is minimal and does not present a new problem. The instructor's voice is heard, but the content is not clearly related to the previous problems. This section seems to be a transition or a summary of the course's structure rather than a new problem-solving demonstration.

  8. 30:00 – 32:33 30:00-32:33

    The video ends with a final, unstructured segment. The instructor is seen in the bottom right corner, and the whiteboard is mostly blank except for the words 'Advanced Roots' and 'Foundation' written in red. There is no new problem presented, and the instructor does not speak. The screen remains static for the final seconds, indicating the end of the lecture. The 'KG' logo is visible in the bottom left corner.

This video provides a structured and methodical tutorial on applying linear equations to solve a variety of word problems. The core teaching strategy is a consistent, step-by-step approach: define variables, translate the problem into an equation, and solve it. The problems progress from simple systems of two linear equations (Q1, Q2) to more complex scenarios involving quadratic equations (Q5, Q7, Q9) and rational equations (Q10). The instructor demonstrates key algebraic techniques such as substitution, elimination, factoring, and the quadratic formula. The lesson covers diverse real-world contexts, including number relationships, geometry (rectangle dimensions), and motion (speed and time), making it a comprehensive guide for students to develop their problem-solving skills in algebra.