Problem 27 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ 0.12(y-6)+0.06 y=0.08 y-0.7 $$

Step-by-Step Solution

Answer

The solution is \(y = 0.2\).

Step 1: Distribute and Simplify

First, distribute the 0.12 across the term \((y-6)\). This gives us \(0.12y - 0.72\). Now the equation becomes \(0.12y - 0.72 + 0.06y = 0.08y - 0.7\).

Step 2: Combine Like Terms

Combine the \(y\) terms on the left side: \(0.12y + 0.06y\) becomes \(0.18y\). The equation is now \(0.18y - 0.72 = 0.08y - 0.7\).

Step 3: Rearrange the Equation

Get all the terms involving \(y\) on one side by subtracting \(0.08y\) from both sides: \(0.18y - 0.08y - 0.72 = -0.7\). This simplifies to \(0.10y - 0.72 = -0.7\).

Step 4: Solve for y

Add \(0.72\) to both sides to isolate the \(y\) term: \(0.10y = -0.7 + 0.72\). Simplify the right side to get \(0.10y = 0.02\).

Step 5: Divide to Find y

Divide both sides by 0.10 to solve for \(y\): \[ y = \frac{0.02}{0.10} = 0.2 \].

Key Concepts

Equation Solving

When tackling algebraic equations, the ultimate goal is to find the value of the variable that satisfies the equation. This process is known as equation solving. To effectively solve equations, follow these guiding steps:

• First, simplify each side of the equation as much as possible.
• Transform the equation to have the variable on one side, making it easier to isolate.
• Perform the same operation on both sides of the equation to keep them equal.
• Finally, isolate the variable to find its value.

Remember to check your solution by plugging the value of the variable back into the original equation to see if it holds true. This verifies that the solution is correct. Equation solving is an essential step in algebra that helps us discover unknown values.

Distributive Property

The distributive property is a key concept in algebra that allows you to simplify expressions. It states that a term outside a parenthesis can be distributed to each term inside a parenthesis. For example:

• If you have an expression: \(a(b + c)\), then it becomes \(ab + ac\) after applying the distributive property.

In the given exercise, this property is used to distribute 0.12 across the term \((y - 6)\). This gives us a new expression \(0.12y - 0.72\). Using the distributive property is helpful because it reduces complex equations into simpler ones, which can then be solved more easily.When you apply the distributive property, it is important to keep the signs of each term within the parenthesis - this ensures accuracy in your solutions.

Combining Like Terms

Once we have simplified expressions using the distributive property, the next step is to combine like terms. Like terms have the same variable raised to the same power, only their coefficients differ.

• In the example equation, after distribution and simplifying, we notice: \(0.12y + 0.06y\) on the left side, which are like terms because they both contain the variable \(y\).

By adding the coefficients of these like terms together, the expression simplifies to \(0.18y\). Combining like terms is crucial to making equations easier to handle and solve. Always keep an eye out for terms that can be combined to reduce clutter and simplify the solving process.

Isolating Variables

Isolating the variable means getting the variable by itself on one side of the equation so we can see what it equals. This step is essential for solving the equation and finding the value of the variable.To isolate the variable, you might need to:

• Move terms with the variable to one side of the equation. This often involves adding or subtracting terms from both sides.
• Cancel out coefficients by dividing both sides of the equation.

In the sample problem, we move \(0.08y\) from the right side to the left to isolate terms containing \(y\). This gave us \(0.10y - 0.72 = -0.7\). Next, we added 0.72 to both sides, resulting in \(0.10y = 0.02\), and finally, divided by 0.10 to isolate \(y\), concluding with \(y = 0.2\). Simplifying equations by isolating variables can seem challenging at first, but with practice, it unlocks the solution to many problems.