Problem 45 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{x-3}{5}-1=\frac{x-5}{4}$$

Step-by-Step Solution

Answer

After resolving the equation, the solution obtained is 'x' = -7, which has been verified to be correct.

Step 1: Remove fractions

Multiply each term of the equation by the least common denominator (LCD) of 5 and 4, which is 20. This should result in the new equation \(20[\frac{x-3}{5}-1] = 20[\frac{x-5}{4}] \), which simplifies to \( 4(x-3) - 20 = 5(x-5) \).

Step 2: Simplify the equation

Distribute the numbers outside the brackets; thus the equation now looks like \( 4x - 12 - 20 = 5x - 25 \). Further simplification gives \( 4x - 32 = 5x - 25 \).

Step 3: Group 'x' terms together and solve for 'x'

Subtract 4x from both sides to isolate 'x' terms, yielding \( -32 = x - 25 \). Then, add 25 to both sides to solve for 'x', resulting in \( x = -7 \).

Step 4: Check the proposed solution

Confirm that 'x' = -7 is indeed the solution by substituting 'x' into the original equation \(\frac{x-3}{5}-1=\frac{x-5}{4}\) and checking if both sides are equal. With 'x' = -7, the left side becomes \(\frac{-7-3}{5}-1 = -3\), and the right side becomes \(\frac{-7-5}{4} = -3\). Since the left side equals the right side, 'x' = -7 is indeed the correct solution.

Key Concepts

Solving Equations

When it comes to solving equations, the primary goal is to find the value of the variable that makes the equation true. This often involves manipulating the equation by adding, subtracting, multiplying, or dividing terms to isolate the variable. In our example, we start with an equation in the form \( \frac{x-3}{5} - 1 = \frac{x-5}{4} \). The fractions in this equation can make it appear complicated, but a great first step is to eliminate the fractions by finding a common denominator or multiplying both sides to simplify.

• In the solution provided, the equation was multiplied by the least common denominator of all terms.
• This transformed the equation into a simpler form without fractions: \(4(x-3) - 20 = 5(x-5)\).
• Solving then becomes a process of expanding terms, combining like terms, and isolating the unknown variable.

By taking these steps, we can solve for \(x\) and determine that \(x = -7\) is the solution.

Least Common Denominator

The least common denominator (LCD) is a useful concept when solving equations that contain fractions. It refers to the smallest number that all of the denominators can divide into without leaving a remainder. In our exercise, we have denominators of 5 and 4. The LCD for these numbers is 20.

• To find the LCD, you can list the multiples of each denominator until you find the smallest common multiple.
• Once you've identified the LCD, multiply each term of the equation by this number to eliminate the fractions.
• This results in an equation that is easier to work with as it no longer contains fractions.
• Multiplying by the LCD is a common technique to simplify equations and is particularly effective when dealing with multiple fractions.

After applying the LCD, the equation becomes straightforward to manipulate and solve.

Check Solutions

Checking your solutions is a critical step in solving equations. Once you've found a potential solution, like \(x = -7\) in our example, substituting it back into the original equation verifies its correctness.

• This ensures that the solution satisfies the original equation.
• Both sides of the equation should equal when you substitute the solution back in.
• In our example, substituting \(x = -7\) into \(\frac{x-3}{5} - 1\) and \(\frac{x-5}{4}\) both yield \(-3\), confirming the solution is correct.
• By checking solutions, you can catch errors early and ensure accuracy in solving mathematical problems.

Always remember, verifying your solution by substitution is a robust way to confirm the accuracy of your results.