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Questions and Answers
What is the first step to solve a linear equation involving fractions?
What does cross-multiplying help with when solving linear equations with fractions?
In linear equations with fractions, why do you need to simplify the expression after eliminating the fractions?
When solving a linear equation involving fractions, what happens when you multiply both sides by the LCM of the denominators?
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What is the final step in solving a linear equation with fractions?
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Why is it important to eliminate fractions while solving linear equations?
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What field uses linear equations with fractions to solve problems related to forces, work, and energy?
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How are linear equations with fractions applied in chemistry?
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In structural engineering, which system involves linear equations with fractions?
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What does the coefficient in front of a chemical formula represent?
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How are linear equations with fractions utilized in mathematics and engineering?
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What is the first step to solve a linear equation involving fractions?
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Why must a variable appearing under a fraction also appear in the denominator when solving linear equations with fractions?
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When manipulating linear equations with fractions, what is the purpose of following properties like transitivity and commutativity?
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In solving linear equations with fractions, why is it necessary to multiply or divide by a constant number to clear fractions?
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After simplifying the linear equation, what is the final step in the process of solving for the variable?
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Study Notes
Linear Equations with Fractions
Linear equations are mathematical expressions of relationships between variables where the highest power of each variable is 1. When these equations involve fractions, they can be solved by manipulating the equation to isolate the fraction and reduce it to its simplest form. Here's how you can do this:
Step 1: Get Rid of Fractions
To solve a linear equation involving fractions, your first step is to eliminate them. This usually involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators of all fractions involved. For example, if you have two fractions (x+1)/(x-1) and (-x+2)/(x+3), their LCM would be (x^2 - x + 2)(x^2 + x + 3). By multiplying both sides of the equation by the LCM, the fractions cancel out.
Step 2: Simplify the Fraction
Once you've eliminated the fractions, you may need to further simplify the resulting expression. To do this, you can cross-multiply or distribute and combine like terms as needed. For instance, if you have ((x+1)*(x-1))/(x^2 - x + 2) = (-x+2)*(x+3)/(x^2 + x + 3), cross-multiplying gives you (x^2 - x + 2)((-x+2)*(x+3)) = ((x+1)*(x-1))*((x^2 + x + 3)). Then, distributing and combining like terms results in -x^3 + 2x^2 - 6x + 6 = x^3 - x^2 - x + 2x - 2. Finally, simplifying the left side gives you -x^3 + x^2 + 6x - 4 = 0.
Step 3: Solve the Equation
The last step is to solve the resulting equation. If it's already written in standard form (ax^n + bx^(n-1) + ... = c), you can set it equal to zero and apply any appropriate algebraic methods such as factoring or completing the square depending on the number of variables and the complexity of the coefficients. In our example above, we now have -x^3 + x^2 + 6x - 4 = 0. We can factor this as -(x+2)(x^2 - 3) = 0, leading to the solutions x=-2 and x=sqrt(3).
In conclusion, solving linear equations with fractions involves getting rid of fractions using the LCM, simplifying the resulting expression through cross-multiplication or distribution, and solving the final equation using algebraic techniques.
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Description
Learn how to solve linear equations involving fractions by eliminating the fractions, simplifying the resulting expression, and applying algebraic methods to find the solutions. Practice manipulating equations with fractions in this quiz.
