Problem 62 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ \frac{5}{9} x+2-\frac{1}{6} x=\frac{11}{18} x+\frac{1}{3} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 7.5\).

Step 1: Isolate Variable Terms

Let's start by getting all the terms that contain the variable \(x\) on one side of the equation. We begin by subtracting \(\frac{11}{18}x\) from both sides: \(\frac{5}{9}x - \frac{1}{6}x - \frac{11}{18}x + 2 = \frac{1}{3}\).

Step 2: Simplify Variable Terms

To combine the terms \(\frac{5}{9}x\), \(-\frac{1}{6}x\), and \(-\frac{11}{18}x\), let's get a common denominator. The least common denominator of 9, 6, and 18 is 18. Therefore, convert each term:\(\frac{5}{9}x = \frac{10}{18}x\), \(-\frac{1}{6}x = -\frac{3}{18}x\), now the equation is \(\frac{10}{18}x - \frac{3}{18}x - \frac{11}{18}x + 2 = \frac{1}{3}\).

Step 3: Combine Like Terms

Combine the coefficients of \(x\): \(\frac{10}{18}x - \frac{3}{18}x - \frac{11}{18}x = \frac{-4}{18}x\). So, the equation becomes: \(-\frac{4}{18}x + 2 = \frac{1}{3}\).

Step 4: Simplify Coefficient

Simplify the coefficient of \(x\) in \(-\frac{4}{18}x\), which reduces to \(-\frac{2}{9}x\). So the equation is now:\(-\frac{2}{9}x + 2 = \frac{1}{3}\).

Step 5: Isolate the Variable

Subtract 2 from both sides to isolate the \(x\)-term:\(-\frac{2}{9}x = \frac{1}{3} - 2\).Convert 2 to \(\frac{6}{3}\) to have a common denominator and subtract:\(-\frac{2}{9}x = \frac{1}{3} - \frac{6}{3} = -\frac{5}{3}\).

Step 6: Solve for x

Multiply both sides by the reciprocal of \(-\frac{2}{9}\) to solve for \(x\):\(x = -\frac{5}{3} \times -\frac{9}{2}\).This simplifies to \(x = \frac{45}{6}\).Simplify further to get \(x = \frac{15}{2}\) or \(x = 7.5\).

Key Concepts

Solving Equations

To solve algebraic equations, we go through a series of logical steps to find the value of the variable that makes the equation true. Imagine it like a puzzle, where you're searching for the pieces that fit perfectly. First, you need to understand what the equation is asking. In our example, we want to find the value of \(x\) that balances both sides of the equation. This involves getting the terms that contain \(x\) together and simplifying wherever necessary.

The key steps in solving an equation are:

• Combining like terms
• Ensuring all terms with variables are on one side
• Clearing fractions by using common denominators
• Simplifying expressions
• Isolating and solving for the variable

Each of these steps helps to progressively simplify the equation until you break it down to the simplest form, revealing the value of \(x\).

Variable Isolation

Variable isolation is all about getting the variable \( x \) alone on one side of the equation. It helps you see clearly what the value of \( x \) should be. Initially, your equation might look cluttered with different terms and numbers all mixed up.

To isolate \( x \):

• Move all terms containing \( x \) to one side; this might involve adding, subtracting, or dividing terms from both sides.
• Once \( x \) is alone with its coefficient, then you can easily solve for it.

In our specific example, we first ensured all the \( x \)-terms were on one side by subtracting \( \frac{11}{18}x \) from both sides. This gave us a neat setup to further simplify and solve the equation.By isolating the variable, we transform our equation into something manageable and easier to interpret.

Simplifying Fractions

Fractions can be tricky, but with the right approach, you can simplify them with ease. Simplifying fractions involves reducing them to their simplest form so that calculations become more straightforward.

Here’s how you do it:

• Find a common factor for the numerator and the denominator.
• Use this factor to divide both the numerator and the denominator.

For example, in the equation, we had the term \(-\frac{4}{18}x\). By dividing 4 and 18 by their greatest common factor of 2, we simplified it to \(-\frac{2}{9}x\). This simplification helps reduce the complexity of the equation making it easier to solve. With practice, you'll get used to spotting opportunities to simplify fractions, and it will save you time in solving equations.

Common Denominator

Getting the least common denominator (LCD) is a crucial step when you’re dealing with equations involving multiple fractions. This process allows you to combine fractions easily and simplifies the equation.

Here's the way to find a common denominator:

• Identify the denominators of all the fractions you are working with.
• Find the least common multiple of these denominators.
• Convert each fraction to an equivalent one that has this common denominator.

In our example, we identified 18 as the LCD for \( \frac{5}{9}x \), \( -\frac{1}{6}x \), and \(-\frac{11}{18}x\). This was the key to writing each term with a denominator of 18, allowing us to perform operations like addition and subtraction with ease.

Once the fractions are under a common denominator, you can proceed to combine them as if they were simple integers which massively simplifies the overall solving process for the equation.