Problem 12 - Algebra Form and Function

Question

Solve the equations. $$ 3 d-\frac{1}{2}(2 d-4)=-\frac{5}{4}(d+4) $$

Step-by-Step Solution

Answer

Answer: The value of d is -28/13.

Step 1: Eliminate the Fractions

To do this, we will find the least common multiple (LCM) of the denominators and multiply each term by the LCM to eliminate the fractions. The denominators are 2 and 4, and their LCM is 4. So, multiplying both sides of the equation by 4, we get: $$ (4)\left[3 d - \frac{1}{2} (2 d - 4) = -\frac{5}{4} (d + 4) \right] $$

Step 2: Distribute the Multiplication

Now we will distribute 4 to each term inside the brackets on both sides of the equation: $$ 12 d - 2 (2 d - 4) = -5 (d + 4) $$

Step 3: Distribute Within Parentheses

Now we have to distribute the -2 and -5 into the terms inside the parentheses: $$ 12 d - 4 d + 8 = -5 d - 20 $$

Step 4: Combine Like Terms

Now, we'll combine the like terms on both sides of the equation: $$ (12 d - 4 d) + 8 = (-5 d) - 20 $$ $$ 8 d + 8 = -5 d - 20 $$

Step 5: Isolate the variable d

First, add 5d to both sides to eliminate d on the right side: $$ 8 d + 5 d + 8 = -5 d + 5 d - 20 $$ $$ 13 d + 8 = - 20 $$ Now, subtract 8 from both sides to eliminate the constant term on the left side: $$ 13d + 8 - 8 = - 20 - 8 $$ $$ 13 d = -28 $$ Finally, divide both sides by 13 to find the value of d: $$ d = \frac{-28}{13} $$ So, the solution to the equation is: $$ d = -\frac{28}{13} $$.

Key Concepts

Fractions

Fractions can often make equation solving a bit tricky. In algebra, they appear quite frequently, especially in complex equations. To simplify solving the equation, we often eliminate fractions by finding a common multiple of the denominators. This step makes it easier to handle calculations because we convert fractions into whole numbers. For example, if you have fractions with denominators 2 and 4, their least common multiple (LCM) is 4. Multiply every term in your equation by this LCM to rid the equation of fractions, resulting in a more straightforward problem to solve.

Steps to eliminate fractions:

• Identify all denominators in the equation.
• Find the least common multiple (LCM) of these denominators.
• Multiply every term in the equation by this LCM.

This technique is crucial because it paves the way for simpler, fraction-free equations, making it easier to proceed with solving.

Distributive Property

The distributive property is a fundamental principle of algebra. It allows you to simplify expressions by distributing or spreading out a multiplier over terms within parentheses. The mathematical form of this property is \(a(b + c) = ab + ac\). This means you multiply the external factor with each term inside the parentheses individually.

In our equation, after multiplying by the LCM, we used distributive property twice. First, apply it to the entire expression to clear fractions, then apply it again inside parentheses to simplify further:

• Apply the multiplication outside the parentheses to each term inside.
• Look inside the parentheses and apply the property again if needed.

Using the distributive property allows breaking down complex expressions into simpler terms, enabling easier manipulation and solution of the equation.

Combining Like Terms

Once an equation is simplified using the distributive property, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power. By adding or subtracting these terms, you simplify the equation further. This step helps in reducing the equation to its simplest form, making it less complex to solve.

For example, in our equation:

• Add/subtract terms with the same variable on one side of the equation.
• Simplified like terms: \(12d - 4d\) became \(8d\).

Combining like terms reduces the clutter in an equation, bringing us one step closer to isolating the variable and finding the solution.

Isolating Variables

The ultimate goal in solving an equation is isolating the variable on one side of the equation. This action enables you to find the value of the unknown. To isolate the variable, you need to perform operations that reverse the effect of what the rest of the equation does to the variable.

In the given equation, we gradually isolate \(d\):

• Move terms containing the variable to one side of the equation.
• Move constant numbers to the opposite side.
• Once the variable is isolated, divide by its coefficient to solve for it.

For example, we did this by first adding \(5d\) to bring all \(d\) terms to one side, then subtracting constants to the other side, and finally dividing by the coefficient of \(d\) to find \(d = \frac{-28}{13}\). This process allows a direct path to the solution of the equation.