Solving Equations with Rational Expressions
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Questions and Answers

به چه معنا‌یی گفته می‌شود که یک عبارت الجبرایی حاوی کسرها است؟

  • حداقل در جمع یا تفریق دو عبارت الجبرایی از بالا و پایین استفاده شده است
  • هم نمای کسر و هم نمای جذر دارد
  • شامل حداقل یک متغیر به توان عددی در صورت یا مخرج است
  • هر دو وجود متغیر در صورت و مخرج دارد (correct)
  • چگونه می‌توان یک عبارت الجبرایی را ساده کرد؟

  • با ضرب کردن صورت و مخرج در عدد گروه بشمارنده
  • با استفاده از رابطه اصلی جبر
  • با جمع یا تفریق اعضای صورت و یا مخرج
  • با تقسیم صورت و مخرج بر عدد اول بزرگترشان (correct)
  • چرا عبارت \( \sqrt{x^2} + y \) به عنوان یک کسر الجبرایی در نظر گرفته نمی‌شود؟

  • حاوی جذر است که در کسرهای الجبرایی نمی‌آید (correct)
  • حاوی توان زوج است که در کسرهای الجبرایی نمی‌آید
  • صورت آن خطی و مخرج نامعلوم است
  • نامعقول است و قابل ساده‌سازی نیست
  • چگونه معادلات حاوی کسرهای الجبرایی را حل کنید؟

    <p>با ساده‌سازی کسرها و سپس جمع یا تفریق آن‌ها</p> Signup and view all the answers

    چند پل به دست خواهید آورد از روش حل معادلات حاوی کسرهای الجبرایی؟

    <p>$2$</p> Signup and view all the answers

    برای حذف کردن کسورات یک معادله، چه باید کرد؟

    <p>ضرب کردن هر دو طرف معادله در عددی که مخرج هر دو کسور مشترک باشد</p> Signup and view all the answers

    چگونه باید اصطلاحات مشابه را ترکیب کنیم تا عبارت ساده‌تر شود؟

    <p>ضرب کردن هر دو طرف معادله در انحصاری کمینه بخشی از مخرج‌ها</p> Signup and view all the answers

    چگونه باید برای حل یک معادله روش صحیح را اعمال کنیم؟

    <p>ضرب کردن همه اصطلاحات تقسیم شده توسط انحصاری کمینه بخشی از مخرج‌ها</p> Signup and view all the answers

    پس از حل یک معادله، چه نکته‌ای باید چک شود؟

    <p>برای رسیدن به جواب، بایستی عبارات جمع و تفاضل گرگامیده‌شده چک شود</p> Signup and view all the answers

    پس از حل یک معادله، چگونه بایستی جواب را ساده‌تر نمایید؟

    <p>پس از حل، سادۀ آن را نشان داد</p> Signup and view all the answers

    Study Notes

    Solving Equations with Rational Expressions

    When working with algebraic expressions involving fractions, which are known as rational expressions, you may encounter situations where these expressions appear within an equation. Solving such equations requires some additional techniques compared to those used with basic algebraic statements. Here's how you can tackle these problems effectively.

    Definition of Rational Expression

    A rational expression is any fraction wherein both the numerator and denominator contain variables raised to powers. For instance, let x and y be real numbers. Then ( \frac{x^2 - x + 1}{y^3} ) is a rational expression, while ( \frac{xy}{2} ) is also a rational expression. However, ( \sqrt{x^2} + y ) is not considered a rational expression because it contains only nonlinear terms.

    Simplifying Rational Expressions Before Solve Equations

    Before we start solving equations containing rational expressions, simplification of these expressions is usually necessary. This involves reducing them into their most basic form. For example, if we have the expression ( \frac{a}{b} ), we could divide both the numerator and the denominator by their greatest common divisor, resulting in ( \frac{\cancel{3}}{\cancel{9}} = \frac{1}{3} ).

    Solving Equations with Rational Expressions

    To solve equations with rational expressions, you can follow these steps:

    1. Clear the fractions: If the equation contains fractions, clear them out by multiplying both sides by a suitable expression. For instance, consider the equation ( \frac{2x}{3} = \frac{x + 1}{2} ). To clear the fractions, we multiply both sides by ( 6 ), which is the product of the denominators of both fractions, as (\frac{2x}{3} \times \frac{1}{1} = \frac{x + 1}{2} \times \frac{3}{3}). This results in ( 2x = 3(x + 1) ), which is a simpler equation involving only integers.

    2. Combine like terms: If the equation contains rational expressions, you should combine like terms to simplify the expression. For example, if you have ( \frac{a}{b} + \frac{c}{d} = \frac{e}{f} ), you can multiply both sides by the least common multiple of the denominators, which in this case is ( bdf ), to get ( adf + bce = ebf ). Now you can solve for the variable, ( x ), in the resulting equation.

    3. Check for extraneous solutions: After solving the equation, you might encounter extraneous solutions, which are solutions that do not satisfy the original equation. For instance, if you have ( \frac{x}{x + 1} = \frac{x - 2}{x - 1} ), you can multiply both sides by ( (x + 1)(x - 1) ) to get ( x(x - 2) = x^2 - x ), which further simplifies to ( x - 2 = 0 ). Solving for ( x ) gives you ( x = 2 ). However, when you check the original equation for this value, you'll find that ( \frac{2}{3} \neq \frac{0}{1} ), so the equation does not hold true for ( x = 2 ).

    4. Simplify the resulting expression: After solving the equation, you might find that the answer involves simplifying fractions. For instance, if you have ( \frac{a}{b} + \frac{c}{d} + \frac{e}{f} = \frac{g}{h} ), you can multiply both sides by the least common multiple of the denominators to get ( ahd + bce + bfg = bgh ), then you can solve for the variable, ( x ), in the resulting equation.

    By following these steps and using the appropriate techniques for solving equations with rational expressions, you can tackle such problems effectively.

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    Description

    Learn how to solve equations involving rational expressions by clearing fractions, combining like terms, checking for extraneous solutions, and simplifying resulting expressions. Master the techniques needed to effectively tackle algebraic problems with rational expressions.

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