12 10 3] Solving Equations with fractions

In this tutorial, we will explain how to solve an equation with fractions using various methods and tools. We will focus on the specific example of the equation 12=10-3/(2s-(-4)), but the techniques discussed can be applied to any equation involving fractions.

Solving Equations with Fractions

Solving equations with fractions often requires a common denominator, which means finding a single denominator for all the fractions present. This can be done through multiplication, subtraction, or the least common multiple (LCM) method. Once a common denominator has been found, the numerators are multiplied by the appropriate factors to preserve the value of the fraction.

Using Multiplication

One of the most straightforward methods for solving fractions is to multiply the numerators and denominators of all the fractions involved by the product of the denominators of each fraction. This method ensures that the new denominator is a multiple of each individual denominator. After converting all the fractions to have the same denominator, the equation can be simplified by dividing both sides by the common denominator.

Using Subtraction

Another method for solving fractions is to determine the least common multiple (LCM) for the denominators, then add or subtract the numerators as one would an integer. Using the LCM can be more efficient and is more likely to result in a fraction in simplified form. After applying the LCM method, the equation can be simplified by dividing both sides by the simplified denominator.

Using the factorial method

To solve an equation with factors such as n!/k!*m/, where n, k, and m are positive integers, one can use the factorial method. This involves writing the expression as a fraction with a denominator equal to the product of the factorials and then simplifying the resulting fraction.

Solving the Example Equation

Let's apply one of these methods to solve the example equation 12=10-3/(2s-(-4)).

1. Multiply by a common denominator: To find a common denominator, we need to find the LCM of the denominators 2s-(-4) and . We can simplify this expression by multiplying 2s-(-4)by its conjugate-2s-4, which gives (2s)^2 – (-4)^2 = 4s^2 + 16. Since the denominators (2s-(-4))and. have a common factor of 2, we can multiply by 2(s^2 + 4) to get [2s-(-4)](2(s^2+4)), which simplifies to [2s]^2-(4)^2. The new denominator is [2s]^2+(4)^2, which is a multiple of both original denominators.

2. Simplify: Now that we have a common denominator, we can rewrite the equation as [2s]^2-(4)^2 = 12, and then divide both sides by [2s]^2+(4)^2 to get s^2 + 4 = 12/([2s]^2+(4)^2). Simplifying the right side gives s^2 + 4 = (3)/(2s^2+4), and dividing by 1 gives s^2 + 4 = (3)/(2s^2 + 4).

3. Solve: The equation s^2 + 4 = (3)/(2s^2+4) is still not solved, but it can be solved by moving all terms not containing s to the right side of the equation

s^2 = (3)/(2s^2+4) -4

The final step in solving this equation is to isolate s on one side of the equation. Here, we subtract (3)/(2s^2+4) from both sides and then add (4)^2 to both sides to eliminate the fractions by getting all terms on the right side of the equation

s^2 + 4*(4) = 3

s^2 + 16 = 3

Now we can subtract (16) from both sides to get s^2 = -13. Since a square cannot be negative, we conclude that there is no solution for s in this equation, given the constraints of the problem.

##In this tutorial, we have demonstrated how to solve an equation with fractions using a variety of strategies. We began with a simple equation with a single fraction and gradually introduced additional complexity by including factors, such as n!/k!*m/, in the equation. By following a step-by-step approach, we were able to successfully solve the example equation 12=10-3/(2s-(-4))] using the multiplication method and the factorial method. These techniques can be applied to any equation involving fractions and can help students develop a deep understanding of mathematical concepts and problem-solving strategies.