Problem 17 - Algebra Form and Function

Question

Solve the equations. $$ \frac{3}{2 x-1}+\frac{5}{3-2 x}=0 $$

Step-by-Step Solution

Answer

Answer: The value of x that satisfies the equation is x = -1.

Step 1: Find the common denominator and eliminate the fractions

To eliminate the fractions, we need to find a common denominator for the two terms. In this case, it is the product of the two denominators, which is \((2x-1)(3-2x)\). We then multiply both sides of the equation by this common denominator. $$ (2x-1)(3-2x)\left(\frac{3}{2 x-1}+\frac{5}{3-2 x}\right)=0(2x-1)(3-2x) $$

Step 2: Simplify the expression

Now that we have eliminated the fractions, let's expand the equation and simplify it: $$ 3(3-2x)+5(2x-1)=0 $$ Expanding and simplifying gives: $$ 9-6x +10x-5=0 $$ Combining like terms: $$ 4x+4=0 $$

Step 3: Solve for x

Now we can solve for x by isolating x on one side: $$ 4x=-4 $$ Divide by 4: $$ x=-1 $$ Therefore, the solution to the given equation is: $$ x=-1 $$

Key Concepts

Solving Equations

When we solve algebraic equations, our goal is to find the value of the variable that makes the equation true. In this problem, the variable is "x". Solving equations often involves several steps: simplifying expressions, finding a common denominator (if fractions are involved), and isolating the variable.
However, applying transformations that keep the equation balanced is key. This means whatever you do to one side, you must also do to the other.

• Start by simplifying each side of the equation.
• Look for a common denominator to eliminate fractions.
• Isolate the variable to solve for it.

Ultimately, it's all about manipulating the equation to make it easier to solve. In this exercise, the variable "x" is isolated by finding a common denominator for the fractions and eliminating them first.

Common Denominator

A common denominator is essential when you have fractions in an equation. Fractions make simple calculations complex, so we use common denominators to combine and eliminate them.
To find a common denominator, identify the least common multiple of the denominators involved. In this exercise, the denominators are \(2x-1\) and \(3-2x\). The common denominator is their product: \( (2x-1)(3-2x) \).

• This product helps simplify the equation because it removes the fractions completely when both sides are multiplied by it.
• Once fractions are gone, you can expand and simplify the remaining expression.

Finding the common denominator is an effective way to unite fractions, making it easier to solve equations in the simplest form possible.

Fractions

Fractions can often make algebraic problems seem harder. However, once you understand how they work, they become much less intimidating. In algebra, fractions occur when the numerator and the denominator involve variables, like in our original exercise.\

• The goal is to eliminate fractions to simplify things.
• Numerators often need to be manipulated to simplify or solve.

In this case, multiplying everything by the common denominator \((2x-1)(3-2x)\) allows you to remove the fractions. This simplifies the entire equation, leaving only integers to work with.Understanding fractions in algebra means grasping how they affect the equation and learning how to effectively simplify and solve them.