Question

Solve the equation. Round the result to the nearest hundredth. $$ -2(4.36-6.92 x)=9.27 x+3.87 $$

Short answer

The solution to the equation, rounded to the nearest hundredth, is \( x = 2.76 \)

Step-by-step solution

Simplify the Equation

Start by simplifying the left side of the equation. Distribute the multiplication across the brackets. This gives: \( -2 \times 4.36 + 2 \times 6.92x = 9.27x + 3.87 \). After multiplying, you have: \( -8.72 + 13.84x = 9.27x + 3.87 \)

Move Terms Involving x to One Side

Next, to isolate x, move all the x terms to one side and the constants to the other. Subtracting \(9.27x \) from both sides yields: \( -8.72 + 4.57x = 3.87 \). And similarly, adding \(8.72\) to both, we get: \( 4.57x = 12.59 \)

Solve for x

Finally, to solve for x, divide both sides of the equation by the coefficient of x, which is 4.57. Thus: \( x = 12.59 / 4.57 \). Using a calculator to divide gives the solution.

Key concepts

Understanding the Distributive Property

When solving linear equations, the distributive property is a tool that allows you to multiply a single term by each term within a set of parentheses. This property is vital for simplifying complex expressions. Let's look at how it's applied in our example.

In the original exercise given, we encounter the expression \( -2(4.36 - 6.92x) \). To simplify this, we use the distributive property. You multiply \( -2 \) by each term inside the parentheses, which gives us \( -2 \times 4.36 \) and \( -2 \times -6.92x \). Consequently, the expression simplifies to \( -8.72 + 13.84x \). It's crucial to keep the signs in mind during this process to avoid common mistakes.

Applying the distributive property correctly can often make the difference between an equation that feels complicated and one that's manageable. It's the first step in transforming the equation into a form where you can start to isolate the variable.

Isolating Variables to Solve Equations

Isolating the variable is a core step in finding the solution to an equation. It involves moving all terms containing the variable to one side and constants to the other side to ultimately have the variable alone on one side of the equation.

In our exercise, after applying the distributive property, we work on isolating \(x\). To do this, we must first deal with the term \(9.27x\) on the right side of the equation. By subtracting \(9.27x\) from both sides, we effectively move all the \(x\)-terms to one side, simplifying the equation to \(4.57x = 3.87 + 8.72\). Now, the variable \(x\) is on the left side with a coefficient of \(4.57\), and we are one step closer to finding its value.

The goal is to have \(x\) completely isolated, meaning no other numbers or variables are attached to it. This makes it much easier to see the solution.

Simplification of Equations

Equation simplification is the process of reducing an equation to its simplest form, making it easier to solve. Simplifying involves combining like terms, using properties of equality, and performing basic arithmetic operations.

In the solution presented for our exercise, once we have isolated the variable to one side as \(4.57x\), the next step is to simplify the right side by adding the constants: \(3.87 + 8.72\) to get \(12.59\). With this, the equation on hand is \(4.57x = 12.59\).

To simplify further and solve for \(x\), we divide both sides by the coefficient of \(x\), which is \(4.57\). This final simplification step gives us \(x = 12.59 / 4.57\), showcasing the variable \(x\) by itself. Through simplification, you're able to transform a complex equation into a straightforward expression that can be easily solved using basic arithmetic.

Remember, always perform operations carefully and check your work at each stage to ensure accuracy. Simplification can reveal the solution more clearly and make it more understandable.