Problem 34 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ \frac{3 x-1}{5}-2=\frac{2-x}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{43}{14}\).

Step 1: Eliminate the fractions

To solve the equation, first eliminate the fractions by finding a common denominator for the fractions on both sides. The common denominator for 5 and 3 is 15. Multiply every term in the equation by 15 to eliminate the fractions:\[15 \times \left(\frac{3x-1}{5}\right) - 15 \times 2 = 15 \times \left(\frac{2-x}{3}\right)\]Simplifying this, we get:\[3(3x-1) - 30 = 5(2-x)\]

Step 2: Distribute the terms

Distribute the coefficients 3 and 5 to the terms inside the parentheses:\[9x - 3 - 30 = 10 - 5x\]Simplifying further, we have:\[9x - 33 = 10 - 5x\]

Step 3: Move all terms involving x to one side

Add \(5x\) to both sides to move all terms involving \(x\) to the left side:\[9x + 5x - 33 = 10\]This simplifies to:\[14x - 33 = 10\]

Step 4: Isolate the term with x

Add 33 to both sides to move the constant term to the right side of the equation:\[14x = 10 + 33\]This simplifies to:\[14x = 43\]

Step 5: Solve for x

Divide both sides by 14 to solve for \(x\):\[x = \frac{43}{14}\]

Step 6: Check your solution

Substitute \(x = \frac{43}{14}\) back into the original equation to verify the solution.- Left Side:\[\frac{3 \times \frac{43}{14} - 1}{5} - 2\]\(= \frac{\frac{129}{14} - 1}{5} - 2\)\(= \frac{\frac{115}{14}}{5} - 2\)\(= \frac{23}{14} - 2\)\(= \frac{23}{14} - \frac{28}{14}\)\(= -\frac{5}{14}\)- Right Side:\[\frac{2-\frac{43}{14}}{3}\]\(= \frac{\frac{28}{14} - \frac{43}{14}}{3}\)\(= \frac{-15}{14} \times \frac{1}{3}\)\(= -\frac{5}{14}\)Since both sides are equal, our solution \(x = \frac{43}{14}\) is correct.

Key Concepts

Fraction Elimination

When you encounter an equation with fractions, eliminating them simplifies your work. To do this, find a common denominator for the fractions involved. In our example, the denominators are 5 and 3. Thus, the least common denominator is 15. Multiply every term of the equation by 15 to clear the fractions:

• For the term \(\frac{3x-1}{5}\), multiplying by 15 simplifies to \(3(3x-1)\).
• The number \( -2 \) becomes \(-30\) since \(15 \times -2 = -30\).
• For \(\frac{2-x}{3}\), multiplying by 15 simplifies to \(5(2-x)\).

Now, the equation is free of fractions, making it much easier to solve as a traditional linear equation.

Equation Checking

After solving for the variable, it's vital to check if your solution is correct. Substituting the value back into the original equation will confirm your results. In our problem, we found \(x = \frac{43}{14}\). Let's verify this solution:

• Substitute \(x = \frac{43}{14}\) into the left side of the original equation.
• Carry out the calculations step-by-step to ensure no discrepancies arise.
  • The left side simplifies to \(-\frac{5}{14}\).
• Do the same for the right side; it should also simplify to \(-\frac{5}{14}\).

Since both sides of the equation equal, \(x = \frac{43}{14}\) is indeed the correct solution.

Common Denominator

Finding a common denominator is crucial when dealing with linear equations involving multiple fractions. This common denominator allows us to rewrite each fraction in terms of the same base, making them easier to handle. Here's how:

• Identify the least common multiple of the denominators. In our case, 5 and 3 become 15.
• Multiply every term by this least common multiple to remove the fractions from the equation.

Once you multiply each term by this common denominator, the equation transforms into a simpler algebraic expression, helping you to proceed more smoothly towards isolating the variable.

Variable Isolation

The next critical step is variable isolation, which involves arranging your equation so the variable is alone on one side. Here's how you can do this efficiently:

• Begin by combining like terms. Move every term containing the variable to one side.
• Add or subtract constants to isolate terms with the variable on one side of the equation.
  • In our case, adding 5x yields \(9x + 5x\), simplifying to \(14x\).
• Finally, divide by the coefficient of the variable to solve for it.

These steps lead to \(x = \frac{43}{14}\), providing a clear and correct solution to your equation. Efficient isolation simplifies the linear equation, making it less daunting to solve.