Problem 29 - College Algebra

Question

contain linear equations with constants in denominators Solve each equation. $$ \frac{x+1}{3}-5-\frac{x+2}{7} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 26\).

Step 1: Make the Denominators the Same

To simplify the equation, the denominators must be the same. Find the least common multiple of 3 and 7, which is 21. Multiply each term by 21 to clear the denominators: \[21*\frac{x+1}{3}-21*5-21*\frac{x+2}{7}\] which simplifies to: \[7(x + 1) - 5*21 - 3(x+2)\]

Step 2: Simplify the equation

Now, simplify the equation and try to get all terms involving x on one side of the equation and constants on the other side. Then, apply the distributive property: \[7x + 7 - 105 - 3x - 6 = 0\] leading to: \[4x - 104 = 0\]

Step 3: Isolate the variable x

Lastly, the equation can be solved for x by adding 104 to both sides: \[4x - 104 + 104 = 104\] This simplifies down to: \[4x = 104\] and then dividing both sides by 4 to isolate x: \[x = \frac{104}{4} = 26\]

Key Concepts

Least Common Multiple

When dealing with equations that have different denominators, finding the least common multiple (LCM) is essential. The LCM helps to simplify the problem by making all denominators the same. In this process, you identify the smallest number that is a multiple of both denominators. For example, in the equation \( \frac{x+1}{3}-\frac{x+2}{7} \), we need to find the LCM of 3 and 7.
The LCM of 3 and 7 is 21, because 21 is the smallest number that both 3 and 7 can divide into without leaving a remainder. By multiplying every term in the equation by 21, we eliminate the fractions. This makes it easier to manipulate the equation, allowing us to focus on simplifying and solving for the variable.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and combine like terms. It states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) \) is equivalent to \( ab + ac \).

In the given equation, after finding the LCM and multiplying through to eliminate fractions, we reach:

• \( 7(x+1) \)
• \( -3(x+2) \)

Applying the distributive property, we can expand these:

• \( 7(x+1) \) becomes \( 7x + 7 \)
• \( -3(x+2) \) becomes \( -3x - 6 \)

This helps us combine terms involving \(x\) and also consolidate the constants. The careful application of the distributive property allows us to simplify the equation into a basic form that we can easily solve.

Isolate the Variable

To solve linear equations, a key goal is to isolate the variable, allowing us to find its value. This process involves several steps that simplify the equation, ensuring that the variable stands alone on one side.

Once the equation is simplified using the least common multiple and distributive property, you work towards keeping the variable \(x\) on one side. In our example, we transform the equation into:

• \( 4x - 104 = 0 \)

To isolate \(x\), we need to get rid of the constant term (-104) attached to it. This is achieved by adding 104 to both sides of the equation, resulting in:

• \( 4x = 104 \)

Finally, by dividing each side by 4, we fully isolate \(x\):

• \( x = \frac{104}{4} = 26 \)

By isolating the variable, we end up with the solution: \(x = 26\). This method is applicable to many types of linear equations, making algebraic solutions manageable and straightforward.