Solving Linear Equations with Fractions

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16 Questions

What is the first step to solve a linear equation involving fractions?

Get rid of the fractions by multiplying both sides by the least common multiple (LCM) of the denominators

What does cross-multiplying help with when solving linear equations with fractions?

Simplifying the resulting expression

In linear equations with fractions, why do you need to simplify the expression after eliminating the fractions?

To ensure accuracy in solving

When solving a linear equation involving fractions, what happens when you multiply both sides by the LCM of the denominators?

The fractions cancel out

What is the final step in solving a linear equation with fractions?

Solve the resulting equation

Why is it important to eliminate fractions while solving linear equations?

Fractions can lead to incorrect solutions

What field uses linear equations with fractions to solve problems related to forces, work, and energy?

Physics

How are linear equations with fractions applied in chemistry?

Calculating amount of reactants and products in chemical reactions

In structural engineering, which system involves linear equations with fractions?

Mass-spring-damper system

What does the coefficient in front of a chemical formula represent?

Ratio of atoms or molecules

How are linear equations with fractions utilized in mathematics and engineering?

For solving problems related to forces, work, and energy

What is the first step to solve a linear equation involving fractions?

Add or subtract the same fraction from each side

Why must a variable appearing under a fraction also appear in the denominator when solving linear equations with fractions?

To isolate the variable

When manipulating linear equations with fractions, what is the purpose of following properties like transitivity and commutativity?

To correctly manipulate the equation

In solving linear equations with fractions, why is it necessary to multiply or divide by a constant number to clear fractions?

To make the equation simpler to work with

After simplifying the linear equation, what is the final step in the process of solving for the variable?

Solve for the variable

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.

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.

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