Problem 36 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(2(4 x+1)=-12+6\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1\).

Step 1: Expand the equation

Start by expanding the left side of the equation: \(2(4x + 1) = 8x + 2\). Thus, the equation becomes \(8x + 2 = -12 + 6\).

Step 2: Simplify the right side

Now, simplify the right side of the equation: \(-12 + 6 = -6\). So, the equation is now \(8x + 2 = -6\).

Step 3: Isolate the variable term

Subtract 2 from both sides to isolate the term with the variable: \(8x + 2 - 2 = -6 - 2\). This simplifies to \(8x = -8\).

Step 4: Solve for the variable

Now, divide both sides of the equation by 8 to solve for \(x\): \(x = \frac{-8}{8}\). This simplifies to \(x = -1\).

Key Concepts

Equation Simplification

Equations in algebra often look a bit complex at first, but simplifying them helps make the solution clearer. Simplification involves making an equation as neat and tidy as possible, and one way to achieve this is by expanding brackets or combining like terms.

For example, in the exercise you have a term like \(2(4x + 1)\). To simplify, you need to distribute the 2 across both the \(4x\) and the 1, leading to \(8x + 2\). This step removes the brackets, making it easier to handle the equation. Simplification also involves reducing constants on either side, such as changing \(-12 + 6\) to \(-6\).

Remember, the goal of simplification is to make the equation less complex by breaking down expanded terms and combining constants. This step sets the stage for further manipulations, making it easier to isolate the variable.

Variable Isolation

In algebra, isolating the variable is one of the most crucial steps to finding the solution to an equation. Variable isolation means you manipulate the equation so that the variable you are solving for, like \(x\), stands alone on one side of the equation, usually the left.

In our example, once you have simplified the equation to \(8x + 2 = -6\), the next task is to ensure \(x\) is by itself. You achieve this by moving other numbers across the equation. For instance, subtract 2 from both sides to cancel out the +2, resulting in \(8x = -8\).

This process might involve using inverse operations, such as adding or subtracting terms from both sides of the equation, or using multiplication or division. Remember, what you do to one side, you must do to the other to maintain balance in the equation. This step is all about rearranging terms to focus solely on the variable.

Solving for x

The final step in solving a linear equation is to actually find the value of the variable. After the variable is isolated, solving for \(x\) becomes a straightforward operation.

In the example, with \(8x = -8\), you need to get \(x\) alone by dividing both sides by 8. This will result in \(x = \frac{-8}{8}\) which simplifies to \(x = -1\).

This step focuses on using basic arithmetic to simplify the equation once the variable is isolated. It's important to correctly perform these arithmetic operations to find the correct solution. Once completed, you have effectively found the value of the unknown variable in the original equation.