Problem 65 - Algebra for College Students

Question

Solve each equation, and check the solution. \(\frac{3 x-1}{4}+\frac{x+3}{6}=3\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Eliminate the fractions

To eliminate the fractions, find a common denominator. The denominators are 4 and 6. The least common multiple of 4 and 6 is 12. Multiply every term by 12 to remove the fractions.

Step 2: Multiply each term by 12

Multiplying each term by 12: \[ 12 \times \frac{3x-1}{4} + 12 \times \frac{x+3}{6} = 12 \times 3 \]Simplify the equation: \[ 3 \times (3x-1) + 2 \times (x+3) = 36 \]

Step 3: Distribute and simplify

Distribute the 3 and 2 in the equation: \[ 9x - 3 + 2x + 6 = 36 \]Combine like terms: \[ 11x + 3 = 36 \]

Step 4: Solve for x

Isolate \(x\) by subtracting 3 from both sides: \[ 11x + 3 - 3 = 36 - 3 \]Which simplifies to: \[ 11x = 33 \]Next, divide both sides by 11: \[ x = \frac{33}{11} = 3 \]

Step 5: Check the solution

Substitute \(x = 3\) back into the original equation to verify: \[ \frac{3(3)-1}{4} + \frac{3+3}{6} = 3 \]Simplify inside the fractions: \[ \frac{9-1}{4} + \frac{6}{6} = 3 \]\[ \frac{8}{4} + 1 = 3 \]Simplify to: \[ 2 + 1 = 3 \] which is true.

Key Concepts

common denominator

When solving equations that include fractions, finding a common denominator simplifies the process by eliminating the fractions. This makes the equation easier to handle. In our exercise, the fractions are \(\frac{3x-1}{4}\) and \(\frac{x+3}{6}\). The denominators are 4 and 6. The Least Common Multiple (LCM) of 4 and 6 is 12.

To eliminate the fractions, we multiply each term in the equation by the common denominator, in this case, 12. This transforms our equation from:

• \(\frac{3 x-1}{4}+\frac{x+3}{6}=3\)
• To: 12 \(\times\) \(\frac{3 x-1}{4}\) + 12 \(\times\) \(\frac{x+3}{6}\) = 12 \(\times 3\)

This results in: 3(3x - 1) + 2(x + 3) = 36, removing all fractions from the equation.

distributive property

After eliminating the fractions, we apply the distributive property. This property indicates that a term outside the parentheses can be distributed to each term inside the parentheses. For example:

• 3 \(\times\) (3x - 1) becomes 9x - 3
• 2 \(\times\) (x + 3) becomes 2x + 6

So, our equation 3(3x - 1) + 2(x + 3) = 36 unfolds into:
9x - 3 + 2x + 6 = 36.

Distributing makes it easier to combine like terms, preparing our equation for isolating the variable in the next step.

isolating the variable

To solve for x, we need to isolate it on one side of the equation. After distribution, our equation is 9x - 3 + 2x + 6 = 36. First, we combine like terms:

• 9x and 2x combine to 11x
• -3 and 6 combine to +3

This simplifies to 11x + 3 = 36. Next, we isolate x by performing the same operation on both sides. Subtract 3 from both sides:

11x + 3 - 3 = 36 - 3, which simplifies to 11x = 33.
Finally, divide both sides by 11:

\(x = \frac{33}{11} = 3\).
So, x is isolated and equals 3.

checking solutions

The last step is checking our solution to ensure it's correct. We substitute x = 3 back into the original equation: \(\frac{3(3)-1}{4}+\frac{3+3}{6}=3\).

First, compute inside the fractions:

• \(\frac{9-1}{4}\)
• \(\frac{6}{6}\)

This simplifies to: 2 + 1 = 3, which is true.

Thus, our solution x = 3 satisfies the original equation. This step is essential because it verifies our work and ensures no mistakes were made.