Problem 35 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(3(2 x+5)=-18+9\)

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Distribute the 3

In the given equation, distribute the 3 on the left-hand side to both terms inside the parentheses: \(3(2x + 5) = 6x + 15\). The new equation is now: \(6x + 15 = -18 + 9\).

Step 2: Simplify the Right-Hand Side

Combine the numbers on the right-hand side of the equation: \(-18 + 9 = -9\). So, the equation simplifies to: \(6x + 15 = -9\).

Step 3: Isolate the Variable Term

Subtract 15 from both sides to isolate the variable term: \(6x + 15 - 15 = -9 - 15\). This simplifies to: \(6x = -24\).

Step 4: Solve for x

Divide both sides by 6 to solve for \(x\): \(x = \frac{-24}{6}\). Simplifying the fraction, we find \(x = -4\).

Key Concepts

Distributive Property

The distributive property is an essential tool in algebra that helps us simplify expressions and solve equations. It involves multiplying a single term by each term inside a set of parentheses.
For example, in the expression \(3(2x + 5)\), the number 3 needs to be multiplied by both 2\(x\) and 5. This gives us:

• \(3 \times 2x = 6x\)
• \(3 \times 5 = 15\)

Putting it together results in \(6x + 15\). This step is crucial because it transforms the original equation into a form that is easier to work with. Once the parentheses are removed, we can proceed with other steps to simplify and solve the equation. Remember, distributing correctly lays the groundwork for isolating the variable later on.

Isolation of Variable

Isolating the variable is like clearing the way to find out what the variable represents. In mathematics, this technique is used to simplify the equation by getting the variable we are solving for on one side of the equation by itself.
In our example, once we have the equation \(6x + 15 = -9\), we need to isolate \(x\) by removing any other numbers from its side. We do this by subtracting 15 from both sides of the equation:

• Subtract 15: \(6x + 15 - 15 = -9 - 15\)

This results in \(6x = -24\), where \(x\) is now closer to being all by itself. By isolating \(x\), we make it easier to solve for the exact number that \(x\) equals.

Simplifying Equations

Once we have used the distributive property, the next step often involves simplifying the equation further. This means combining like terms and simplifying both sides of the equation until it is as straightforward as possible.
In the equation \(6x + 15 = -9\), we first simplify the right-hand side. Combining like terms on that side gives us \(-18 + 9 = -9\).

• Rewritten equation: \(6x + 15 = -9\)

The goal here is to keep the equation balanced while making it easier to identify and isolate the variable in the following steps. Simplification helps to reduce any complex terms to a form that's easily manageable and understandable, setting the stage for the final step of solving for the variable.

Solving for x

The last step in solving a linear equation is to solve for \(x\), the unknown variable we are trying to figure out. This involves performing operations that will make \(x\) stand alone on one side of the equation with its coefficient reduced to 1.
In the simplified equation \(6x = -24\), divide both sides by 6:

• \(x = \frac{-24}{6}\)

This simplifies to \(x = -4\).
Here, division is used to isolate \(x\) completely. Solving for \(x\) confirms what the variable is equal to once the other operations (distribution, combination, isolation) are complete. Make sure to check your solution by substituting back into the original equation to verify accuracy, ensuring that the solution makes the equation true.