Problem 130 - Precalculus Essentials

Question

Solve and check: \(\frac{x-3}{5}-\frac{x-4}{2}=5\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x-3}{5}-\frac{x-4}{2}=5\) is \(x=-12\).

Step 1: Make the denominators the same

To solve an equation with fractions, it's easier to work with whole numbers. One way to get rid of the fractions is to make the denominators the same. In this case, the least common denominator (LCD) between 5 and 2 is 10. Multiply each term by 10 to cancel out the fractions: \(10*\frac{x-3}{5}=2*(x-3)\) and \(10*\frac{x-4}{2}=5*(x-4)\). So the equation becomes: \(2*(x-3)-5*(x-4)=50\).

Step 2: Simplify the equation

Next, distribute the numbers outside the parentheses: \(2x-6-5x+20=50\). This simplifies the equation to: \(-3x+14=50\).

Step 3: Isolate the variable

To solve for \(x\), first subtract 14 from both sides to isolate the terms with \(x\), resulting in: \(-3x=36\). Lastly, divide both sides by -3 to solve for \(x\), the equation becomes: \(x=-12\).

Step 4: Check the answer

Substitute \(x=-12\) back into original equation \(\frac{x-3}{5}-\frac{x-4}{2}=5\). After doing the substitution, we get: \(\frac{-12-3}{5}-\frac{-12-4}{2}=5\). This simplifies to: \(\frac{-15}{5}-\frac{-16}{2} = -3+8 =5\). Since both sides equal to each other, this confirms that \(x=-12\) is the correct solution

Key Concepts

Finding the Least Common Denominator

When solving equations with fractions, finding the least common denominator (LCD) is a crucial step. Fractions are easier to manage when they share the same denominator because it enables you to combine them or eliminate them altogether. In our example, the fractions have denominators of 5 and 2. The least common denominator between these two numbers is 10.

• Multiply each term by the LCD to clear the fractions:
  • For \( \frac{x-3}{5} \), multiply by 10, which results in \( 2(x-3) \).
  • For \( \frac{x-4}{2} \), multiply by 10, which results in \( 5(x-4) \).

After multiplying each term by the LCD, you're left with an equation that no longer contains fractions: \(2(x-3) - 5(x-4) = 50\). This simplification transforms the original complex-looking equation into a simpler one that only involves whole numbers.

Understanding the Distributive Property

The distributive property is a key algebraic principle that's useful in many math problems, including our equation. It states that you can multiply a number by a sum by distributing the multiplication across each term inside the parentheses. In our problem:

• We have the expression \(2(x-3) - 5(x-4)\).
  • Distribute \(2\) into \(x - 3\), which becomes \(2x - 6\).
  • Then distribute \(-5\) into \(x - 4\), resulting in \(-5x + 20\).

The distributive property allows you to break down expressions like \((bx-c)\) or \((dx-e)\) and simplify them into more manageable terms. After distributing, the equation becomes \(-3x + 14 = 50\), a much simpler form that's ready to be solved.

Checking Solutions Effectively

Once you solve an equation, it's vital to verify your solution to ensure it's correct. In our problem, after solving for \(x\), we found \(x = -12\). To check this solution, substitute \(-12\) back into the original equation: \(\frac{x-3}{5} - \frac{x-4}{2} = 5\).

• Substitute \(-12\) into the equation:
  • For \(\frac{x-3}{5}\), it becomes \(\frac{-12-3}{5} = \frac{-15}{5} = -3\).
  • For \(\frac{x-4}{2}\), it becomes \(\frac{-12-4}{2} = \frac{-16}{2} = -8\).

Combine these results: \(-3 - (-8) = -3 + 8 = 5\). Since both sides of the equation equal 5, the solution \(x = -12\) is indeed correct. Checking your solution helps you confirm that no mistakes were made during the calculations.