Question

Solve the equation \(5 /(\mathrm{x}-1)+1(4-3 \mathrm{x})=3 /(6 \mathrm{x}-8)\).

Short answer

The solution to the equation \(\frac{5}{x-1}+1(4-3x) = \frac{3}{6x-8}\) is \(x = \frac{-43}{42}\) or \(x=\frac{43}{-6}\).

Step-by-step solution

Rewrite the equation with proper LaTeX syntax

Rewrite the given equation with proper LaTeX syntax: \( \frac{5}{x-1}+1(4-3x) = \frac{3}{6x-8} \)

Determine the LCM of the denominators

Find the least common multiple (LCM) of the denominators, which are \( x-1 \) and \( 6x-8 \). In this case, the LCM is simply the product of the denominators, or: LCM = \((x-1)(6x-8)\)

Multiply both sides of the equation by the LCM

Multiply both sides of the given equation by the LCM: \((x-1)(6x-8) \left (\frac{5}{x-1} + 1(4-3x)\right) = (x-1)(6x-8)\left(\frac{3}{6x-8}\right)\)

Simplify the equation

Now, simplify the equation by cancelling the denominators: \((6x-8) (5) + (x-1)(6x-8)(4-3x) = (x-1)(3)\) Expand the equation: \(30x - 40 + (4x^2 -8x - 18x^2 + 3x(6x-8)) = 3x - 3\) Combine like terms: \(16x^2 - 3x(6x-8) = 3x - 43\)

Solve for x

Now, rearrange the equation and solve for x. First, move all terms to the left side of the equation: \(16x^2 - 3x(6x-8) - 3x + 43 = 0\) Next, factor out a 3x: \(3x(8 - 2(6x-8)) + 43 = 0\) Solve for x: \(x = \frac{-43}{3(14)}\) or \( x = \frac{43}{3(6-8)}\) Thus, the solution is: \(x = \frac{-43}{42}\) or \(x = \frac{43}{-6}\)

Key concepts

Least Common Multiple

When solving rational equations, finding the least common multiple (LCM) of the denominators is a crucial step. The LCM allows us to eliminate the fractions and work with a simpler equation. To calculate the LCM, you need to identify the least common expression that can be a multiple of each denominator's term.
For example, in the equation \( \frac{5}{x-1} + 1(4-3x) = \frac{3}{6x-8} \), the denominators are \(x-1\) and \(6x-8\). To find the LCM, we multiply them together, resulting in \((x-1)(6x-8)\). This LCM makes it easier to manage the equation by getting rid of the fractions when we multiply both sides by it.

• Identify the different factors in each denominator.
• Find the smallest expression that is a multiple of each factor.
• Ensure that you include all unique factors in the LCM.

This process ensures that the LCM encompasses both denominators, creating an expression that will allow for the elimination of fractions.

Simplifying Equations

After multiplying both sides of a rational equation by the LCM, the next task is to simplify the equation. Simplification involves cancelling the fractions by getting rid of the denominators. This is one of the most satisfying parts because the equation looks much cleaner!
In this example, once you multiply the original equation by the LCM \((x-1)(6x-8)\), you're left with expressions like \((6x-8)(5) + (x-1)(6x-8)(4-3x) = (x-1)(3)\).

Here's what you need to do to simplify:

• Cancel out the denominators that you initially dealt with by multiplying by the LCM.
• Eliminate any terms that appear on both sides of the equation similarly.
• Simplify the remaining expression as much as possible to prepare for further solving.

Simplifying makes the equation manageable and sets the stage for expanding and combining like terms.

Expanding Equations

Expansion is a key step in manipulating an equation to reveal its true complexity. You need to distribute terms across sums or differences. This means you take each term in the first parenthesis and multiply it with each term in the second parenthesis. Expansion can oftentimes look daunting, but breaking it down step by step can make it easier.
For the equation given, you encounter an expression like: \((4x^2 - 8x - 18x^2 + 3x(6x-8))\). Here's how to handle the expansion:

• Distribute each term across all others within the parentheses consistently.
• Write out each term from the multiplication so you don't miss anything.
• Simplify the result by combining terms whenever possible in preparation for the next step.

Expanding helps to unlock the internal structure of an equation, making it easier to simplify further.

Combining Like Terms

Once you've expanded the equation, combining like terms gives you a neat and tidy expression. Like terms refer to those terms in an equation that have the exact same variable part, raised to the same power.
For instance, in the expression \(16x^2 - 3x(6x-8) = 3x - 43\), you first need to make sure that you've expanded every part properly. Then, look for terms that share the same variable combination.
Here’s how to combine like terms:

• Identify terms with the same base and exponent.
• Add or subtract their coefficients.
• Write them as a single term.

Combining like terms helps to reduce the equation to its simplest form, making it easier to see the solution. It prepares the equation for any final algebraic solving steps, guiding you toward finding the variable's value efficiently.