Problem 15 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ \frac{5}{6} x+\frac{1}{6} x=-9 $$

Step-by-Step Solution

Answer

The solution is \( x = -9 \).

Step 1: Combine Like Terms

First, simplify the equation by combining the like terms on the left side of the equation. The equation \( \frac{5}{6}x + \frac{1}{6}x = -9 \) can be simplified by adding the coefficients of \( x \):\[ \frac{5}{6}x + \frac{1}{6}x = \frac{6}{6}x = x \]. So, the simplified equation is \( x = -9 \).

Step 2: Solve for x

Since the equation is now \( x = -9 \), we see that the solution is immediate. There is no further calculation needed as we have isolated the variable \( x \) on one side of the equation.

Step 3: Verify the Solution

Substitute \( x = -9 \) back into the original equation to ensure the solution is correct. The original equation is \( \frac{5}{6}x + \frac{1}{6}x = -9 \). Substitute \( -9 \) for \( x \):\[ \frac{5}{6}(-9) + \frac{1}{6}(-9) = -9 \]. Calculate each term: \[ (\frac{5}{6} \times -9) = -\frac{45}{6} = -7.5 \] and \[ (\frac{1}{6} \times -9) = -\frac{9}{6} = -1.5 \]. Adding these terms gives \( -7.5 + -1.5 = -9 \), which matches the right-hand side of the equation, confirming the solution is correct.

Key Concepts

Combining Like Terms

When we solve linear equations, one of the first steps often involves combining like terms. This means merging terms in the equation that have the same variable raised to the same power. For example, in the equation \( \frac{5}{6}x + \frac{1}{6}x = -9 \), both terms are like terms because they each contain the variable \( x \). To combine them, simply add their coefficients:

• Add \( \frac{5}{6} \) and \( \frac{1}{6} \).
• This gives us \( \frac{6}{6}x \), which simplifies to \( x \).

Combining like terms helps simplify complex equations, making them easier to solve. By reducing the number of terms, we can more clearly see the relationship between the variables. This is why combining like terms is a crucial step in solving equations.

Verifying Solutions

After finding a solution to an equation, it's important to verify, or check, that the solution is correct. Verifying a solution ensures no mistakes were made in the initial calculations. To verify the solution of an equation:1. Substitute the solution back into the original equation.2. Perform the calculations to see if both sides of the equation are equal.Let's consider our exercise's solution, \( x = -9 \).

• Substitute \( -9 \) for \( x \) in the original equation: \( \frac{5}{6}(-9) + \frac{1}{6}(-9) \).
• Calculate each term individually: \( -\frac{45}{6} = -7.5 \) and \( -\frac{9}{6} = -1.5 \).
• Add the values: \( -7.5 + -1.5 = -9 \).

The verified calculations match the original equation’s right side, confirming \( x = -9 \) is correct. Verifying solutions not only boosts confidence in the solution but also helps to reinforce understanding of the solving process.

Simplifying Expressions

Simplifying expressions is a key part of solving equations efficiently and effectively. It involves reducing an expression to its most streamlined form by performing basic operations like combining terms and reducing fractions. In our equation \( \frac{5}{6}x + \frac{1}{6}x = -9 \), simplification played a pivotal role. Here's how it works:

• Identify terms that can be combined, such as \( \frac{5}{6}x \) and \( \frac{1}{6}x \), which are like terms.
• Combine these by adding the numerical coefficients to simplify \( \frac{6}{6}x \) to \( x \).
• Always look to simplify fractions during calculations, as this often simplifies the problem significantly.

Simplifying expressions removes unnecessary complexity from equations and often results in direct solutions. It not only prepares the equation for easier manipulation but also makes subsequent problem-solving steps more straightforward.