Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \(x=\frac{37}{3}\). However, upon checking the solution in the original equation, it was discovered that the solution is incorrect.

Step 1: Removing Fractions

To remove the fractions, we multiply every term by the common multiple of the denominators. In this case, 20 is a common multiple for 5 and 4. So the equation becomes \(20*\frac{x-3}{5}-20*1=20*\frac{x-5}{4}\) or \(4(x-3)-20=x-5\).

Step 2: Simplify the Equation

Distribute the 4 to simplify the left side, and do the subtraction on the right side, it results in \(4x-12-20=x-5\), then simplify the left side further to get \(4x-32=x-5\).

Step 3: Grouping terms

Next is to bring similar terms together i.e move all terms in x to one side of the equation and the constants to the other side. This implies : 4x - x = 5 + 32 or \(3x=37\)

Step 4: Final step: Solve for x

To solve for x, divide both sides by 3. This implies: \(x=\frac{37}{3}\)

Step 5: Check the solution

Substitute x = \(\frac{37}{3}\) into the original equation to verify if it is a valid solution. \(\frac{\frac{37}{3}-3}{5}-1 = \frac{\frac{37}{3}-5}{4}\) which simplifies to \(\frac{\frac{37-9}{3}}{5}-1 = \frac{\frac{37-15}{3}}{4}\) and that simplifies to \(\frac{\frac{28}{3}}{5}-1 = \frac{\frac{22}{3}}{4}\) which further simplifies to \(\frac{28}{15}-1 = \frac{22}{12}\). Simplifying the left side and right side of the equation yields \(\frac{13}{15} = \frac{11}{6}\) which is not equal, hence the proposed solution is incorrect.

Key Concepts

Fractions in Equations

Equations involving fractions can often appear intimidating at first glance. However, they can be tackled with ease by eliminating the fractions. To do this, you multiply every term in the equation by the least common multiple (LCM) of the denominators. This step effectively transforms the equation into one without fractions, simplifying the solving process.

For example, in the equation \( \frac{x-3}{5}-1=\frac{x-5}{4} \), the denominators are 5 and 4. The LCM of 5 and 4 is 20. By multiplying each term by 20, you eliminate the fractions, resulting in:

• \( 20 \times \frac{x-3}{5} - 20 \times 1 = 20 \times \frac{x-5}{4} \)
• Which simplifies further to \( 4(x-3) - 20 = 5(x-5) \)

Now, the equation is easier to handle and you can proceed to solve it without the complexities introduced by fractions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in solving equations. It involves reducing expressions to their simplest form, making the equation more straightforward to solve. This usually requires distributing multiplication over addition or subtraction, combining like terms, and performing arithmetic operations.

Let's look at the expression from our equation: \( 4(x-3) - 20 = x - 5 \). The process involves two main steps:

• Distribution: Apply the distributive property to expand \( 4(x-3) \) to \( 4x - 12 \).
• Combine Like Terms: Simplify by performing arithmetic: \( 4x - 12 - 20 \) simplifies to \( 4x - 32 \).

The simplified equation is \( 4x - 32 = x - 5 \), making it easier to isolate the variable and solve for \( x \). By consistently simplifying the expressions, you maintain an organized approach to arriving at the solution.

Verification of Solutions

Verification is the last but essential step in solving equations to ensure the correctness of the proposed solution. It involves substituting the obtained solution back into the original equation to check if both sides are equal.

During this process, any inconsistency indicates an error in the earlier steps. In our solved equation, substituting \( x = \frac{37}{3} \) into the original equation did not satisfy the equation, indicating the need to re-evaluate the solution.

In checking your answers:

• Always substitute the solution into the original equation to verify its correctness.
• Simplify the resulting expressions to ascertain whether both sides of the equation remain balanced.
• If they are not equal, recheck your steps or consider alternative methods for solving the equation.

Verification is valuable as it confirms the accuracy of the solution or highlights the need to revisit your calculations.