Problem 1 - Algebra A Combined Function

Question

Solve each equation. See Examples 1 and \(2 .\) $$ -4 y+10=-2(3 y+1) $$

Step-by-Step Solution

Answer

The solution is \(y = -6\).

Step 1: Simplify the Right Side

First, apply the distributive property to the right side of the equation \(-2(3y + 1)\):\[-2(3y + 1) = -2 \cdot 3y + (-2) \cdot 1 = -6y - 2.\]So, the equation becomes:\(-4y + 10 = -6y - 2\).

Step 2: Move All Terms Involving y to One Side

Next, add \(6y\) to both sides to move all the \(y\) terms to one side: \(-4y + 6y + 10 = -6y + 6y - 2\). This simplifies to:\(2y + 10 = -2\).

Step 3: Isolate the y-Term

Now, subtract \(10\) from both sides to isolate the \(y\)-term: \(2y + 10 - 10 = -2 - 10\). This simplifies to:\(2y = -12\).

Step 4: Solve for y

Finally, divide both sides by \(2\) to solve for \(y\): \(\frac{2y}{2} = \frac{-12}{2}\). Thus, \(y = -6\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept that helps in simplifying expressions and solving equations. It allows you to multiply a single term across terms inside a parenthesis.
In mathematical terms, it can be described by the formula:

• \( a(b + c) = ab + ac \)

This property is particularly useful in solving linear equations where multiple terms are grouped together in parentheses.
In our original exercise, we applied the distributive property to the expression \(-2(3y + 1)\). First, we multiply \(-2\) by \(3y\), and then by \(1\) resulting in \(-6y - 2\).
This simplification is crucial as it sets the groundwork for easier manipulation of the equation in the following steps.

Isolating Variables

Isolating variables refers to the process of rearranging an equation so that the variable you want to solve for is on one side by itself. This is a key step in solving linear equations.
In our example, after applying the distributive property, we have the equation:

• \(-4y + 10 = -6y - 2\)

The next step is to get all terms involving \(y\) on one side of the equation. To do this, we add \(6y\) to both sides, which cancels out the \(-6y\) on the right side, simplifying the equation to \( 2y + 10 = -2 \).
It’s important to work through this step carefully, as any miscalculations could lead to an incorrect solution.

Solving Equations Step-by-Step

Solving equations step-by-step involves breaking down the solution process into manageable parts to ensure each part of the equation is addressed correctly.
After isolating the variable, the final steps involve straightforward arithmetic operations to solve for the variable completely.
In the final steps of our equation:

• Subtract 10 from both sides to simplify the left side to \(2y = -12\).
• Next, divide both sides of the equation by 2 to solve for \(y\).
• This results in \(y = -6\).

By following each of these steps methodically, we ensure that we reach an accurate and verified solution. Solving step-by-step also helps further understanding of the equation-solving process, making it easier to tackle more complex problems in the future.