Problem 26 - Algebra and Trigonometry

Question

Contain linear equations with constants in denominators. Solve each equation. $$\frac{x+1}{4}=\frac{1}{6}+\frac{2-x}{3}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 1 \)

Step 1: Eliminate the Fractions

To get rid of fractions, find the least common denominator (LCD). The denominators here are 4, 6 and 3. So, the LCD is \(12\). Multiply each term of the equation by \$12\$. This gives: \(12*(\frac{x+1}{4})=12*(\frac{1}{6})+12*(\frac{2-x}{3})\), simplifying, it results in \(3(x+1)=2+4(2-x)\).

Step 2: Simplify the Equation

Once the denominators have been eliminated, simplify the equation. After distribution, this gives: \(3x + 3 = 2 + 8 - 4x\). Then simplify again to get: \(3x + 3 = 10 - 4x\)

Step 3: Rearrange the Equation to Find \(x\)

Rearrange the equation to isolate \(x\). First, combine like terms by adding \(4x\) to each side, resulting in \(7x + 3 = 10\). Then, subtract 3 from each side to solve for \(x\): \(7x = 7\). Finally, divide each side by 7 to find that \(x = 1\).

Key Concepts

Least Common Denominator

When dealing with equations that include fractions, like the one in our exercise, the first step is to eliminate the fractions by finding the least common denominator (LCD). In this case, the denominators are 4, 6, and 3. The LCD is the smallest number that all these denominators can divide into evenly, which is 12.
By multiplying each part of the equation by 12, you make the denominators disappear. This process results in the equation:

• From \(\frac{x+1}{4}\) × 12, we get \(3(x+1)\)
• From \(\frac{1}{6}\) × 12, we get 2
• From \(\frac{2-x}{3}\) × 12, we get \(4(2-x)\)

After clearing fractions, you have a simpler equation to work with, making it much easier to solve.

Solving Equations

Solving an equation means finding the value of the variable that makes the equation true. After clearing the fractions, you simplify the equation to see more clearly how to solve for \(x\). In our example, after multiplying out the terms and simplifying, you get \(3x + 3 = 10 - 4x\).
Next, you need to gather all \(x\)-terms on one side of the equation. This involves adding \(4x\) to both sides to eliminate the \(-4x\) on one side, giving you:

• Equation becomes \(7x + 3 = 10\).

Now, move the constant terms to the opposite side. Subtract 3 from both sides to isolate terms with \(x\) on one side:

• Equation becomes \(7x = 7\).

Finally, divide both sides by 7 to solve for \(x\), giving the result \(x = 1\).

Simplifying Expressions

Simplifying expressions is the process of making an equation or term easier to work with. After removing the fractions in our solved equation, we use the distributive property to unlock and combine the terms. Start by multiplying through any brackets:

• We have \(3(x+1)\) which expands to \(3x + 3\).
• We also simplify \(4(2-x)\) to get \(8 - 4x\).

After this, you collect similar terms on either side of the equation which makes it simpler to handle. As shown in our solution, once simplified, the expression \(3x + 3\) combined with \(8 - 4x\) reduces to \(3x + 3 = 10 - 4x\). This operation using simplification aids in solving the equation more efficiently.

Distributive Property

The distributive property is a crucial step in solving equations, especially when they contain brackets. It states that when you multiply a sum by a number, it is the same as multiplying each added part separately and then adding the products.
In our example, the expression \(3(x+1)\) needs distribution before the equation can be solved. Here's how it works:

• Multiply 3 by each term inside the parentheses: \(3 \times x + 3 \times 1\).
• This expands to \(3x + 3\).

Similarly, apply distribution to the term \(4(2-x)\):

• Multiply 4 by 2, and then 4 by \(-x\).
• You get \(8 - 4x\).

By using the distributive property correctly, you break down complex expressions into simpler parts, making the overall equation easier to manage and solve.