Problem 26 - Algebra for College Students

Question

Solve each equation. \(\frac{3 x+1}{9}+2=\frac{x-1}{4}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{-85}{3}\).

Step 1: Clear the Fractions

To eliminate the fractions, find the least common multiple (LCM) of the denominators, which are 9 and 4. The LCM is 36. Multiply every term in the equation by 36 to clear the fractions: \[ 36 \left(\frac{3x+1}{9}\right) + 36(2) = 36 \left(\frac{x-1}{4}\right) \]

Step 2: Simplify the Equation

Simplify each term after multiplication by 36:\[ 4(3x+1) + 72 = 9(x-1) \]Distribute the constants in both sides:\[ 12x + 4 + 72 = 9x - 9 \]

Step 3: Combine Like Terms

Combine and simplify the constant terms on the left side:\[ 12x + 76 = 9x - 9 \]

Step 4: Isolate the Variable

To isolate the variable on one side, subtract 9x from both sides:\[ 12x - 9x + 76 = -9 \]Which simplifies to:\[ 3x + 76 = -9 \]Then, subtract 76 from both sides:\[ 3x = -9 - 76 \]\[ 3x = -85 \]

Step 5: Solve for x

Divide both sides by 3 to solve for \(x\):\[ x = \frac{-85}{3} \]

Key Concepts

Eliminating Fractions

Fractions in equations can sometimes make solving more complex. To simplify, it's a great idea to eliminate them. We do this by finding the least common multiple (LCM) of the denominators. In our problem, the denominators are 9 and 4, and the LCM is 36. By multiplying each part of the equation by 36, the fractions vanish. This process transforms our original equation into a simpler form without those pesky fractions. By using the LCM to "cancel out" fractions, you transform the equation into a format that's easier to handle later on.

Distributive Property

Once fractions are eliminated, use the distributive property to simplify the equation further. This property involves multiplying a single term by each term within a parenthesis. In our example, after multiplying each term by 36, we have:

• 4 times \(3x+1\)
• and 9 times \(x-1\)

Distributing gives us:

• \(4 imes 3x + 4 imes 1\)
• \(9 imes x - 9 imes 1\)

This results in a straight-forward expression without fractions where we can clearly see all terms involved.

Combining Like Terms

After distributing, the goal is to simplify the equation by combining like terms. Like terms are terms that contain the same variables. In the equation, group terms with the variable \(x\) together and constants together. For our problem, after using the distributive property, the left side becomes \(12x + 4 + 72\). We can combine \(4 + 72\) to get \(76\). Thus, we reduce the equation to have:

• Variable terms: \(12x\) on the left and \(9x\) on the right
• Constant terms: \(76\) on the left and \(-9\) on the right

This step lays the groundwork for getting all the variable terms on one side and constant terms on the other.

Isolating the Variable

Once you have simplified as much as possible, the next step is isolating the variable \(x\). You want \(x\) to be on one side of the equation by itself, which involves moving terms across the equation:

• First, subtract \(9x\) from both sides, resulting in \(12x - 9x = 3x\).
• Then, subtract 76 from both sides to move constants, yielding \(3x = -85\).

The final step is to solve for \(x\) by dividing both sides by 3, getting \(x = \frac{-85}{3}\). By performing these operations, we isolate \(x\) making it simple to identify its value.