Problem 13 - Prealgebra

Question

Solve the equation. \(7.3 x-8.9-8.34 x=2.8\)

Step-by-Step Solution

Answer

The solution is approximately \(x \approx -11.25\).

Step 1: Combine Like Terms

The equation given is \(7.3x - 8.9 - 8.34x = 2.8\). First, we combine the like terms on the left side, which are \(7.3x\) and \(-8.34x\). This gives us \((7.3 - 8.34)x\). Calculating this, we have \(-1.04x\). So, the equation becomes \(-1.04x - 8.9 = 2.8\).

Step 2: Isolate the Variable Term

Next, we need to isolate the term containing \(x\) by moving the constant \(-8.9\) to the other side of the equation. Add \(8.9\) to both sides to do this: \(-1.04x - 8.9 + 8.9 = 2.8 + 8.9\). Simplifying, we get \(-1.04x = 11.7\).

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by \(-1.04\). This gives \(x = \frac{11.7}{-1.04}\). Calculating this division results in \(x \approx -11.25\).

Key Concepts

Combining Like Terms

When solving linear equations, a crucial step is combining like terms to simplify the equation. Like terms are terms within the equation that contain the same variable raised to the same power. In the equation \(7.3x - 8.9 - 8.34x = 2.8\), the like terms are \(7.3x\) and \(-8.34x\). Both terms contain \(x\), making them like terms.

Combining these means you perform the operation indicated between them (in this case, subtraction). So, you calculate \(7.3 - 8.34\), which equals \(-1.04\). This transforms the equation into \(-1.04x - 8.9 = 2.8\).

By simplifying the equation, it becomes easier to handle, allowing you to focus on fewer terms. Remember, the key to combining like terms is identifying and grouping terms with the exact same variables and exponents.

Isolating the Variable

After combining like terms, the next step is to isolate the variable. This means you want to get \(x\) by itself on one side of the equation to solve for it easily. In our equation \(-1.04x - 8.9 = 2.8\), the term \(-8.9\) is a constant and needs to be removed from the left side.

You can do this by performing the inverse operation. Since \(-8.9\) is subtracted, you add \(8.9\) to both sides of the equation. This step gives you \(-1.04x - 8.9 + 8.9 = 2.8 + 8.9\).

• On the left side, the \(-8.9\) and \(+8.9\) cancel each other out.
• On the right side, you add \(2.8\) and \(8.9\) to get \(11.7\).

Now, the equation is simplified further to \(-1.04x = 11.7\). By isolating the variable, you're one step closer to finding its value.

Solving for x

The ultimate goal when working with linear equations is to solve for the variable \(x\). Now that we have \(-1.04x = 11.7\), you need to divide by \(-1.04\) to completely isolate \(x\). This step involves performing the division \(x = \frac{11.7}{-1.04}\).

Here's the calculation:

• The division results in approximately \(-11.25\).
• Thus, the value of \(x\) is \(-11.25\).

By dividing both sides by the coefficient of \(x\), you ensure the equation is balanced. It's essential to remember that whatever you do to one side of the equation, you must also do to the other to maintain its equality. Now, you've successfully solved for \(x\) in the equation.