Problem 33 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. $$ 4.65 x-4.79=13.57-6.84 x $$

Step-by-Step Solution

Answer

After carrying out the division and rounding off to the nearest hundredth, the approximate value of \(x\) should be 1.60.

Step 1: Combine like terms

First, combine the terms that involve \(x\) and get them to one side of the equation by adding \(6.84x\) to both sides of the equation: \(4.65x + 6.84x = 13.57 + 4.79\).

Step 2: Simplify

Upon simplifying, we get: \(11.49x = 18.36\).

Step 3: Solve for \(x\)

Solve the equation for \(x\) by dividing both sides by 11.49: \(x = 18.36 / 11.49\).

Step 4: Round to the nearest hundredth

Upon dividing, you will obtain the answer for \(x\). Make sure to round off the answer to the nearest hundredth.

Key Concepts

Linear Equations

Linear equations are algebraic expressions that show the relationship between variables. They have only the first power, meaning terms like \(x^2\) or \(y^3\) don't appear. Linear equations usually look like \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.

In the given equation:

• We have two expressions involving the variable \(x\): \(4.65x - 4.79\) and \(13.57 - 6.84x\).

The goal is to solve for \(x\) by finding the value of \(x\) that makes both sides equal. To do this, we need to get all \(x\) terms on one side of the equation and constant numbers on the other. Understanding these steps is crucial in algebraically manipulating equations correctly.

Combining like terms is a key part of solving linear equations. This means gathering together all terms that contain \(x\) to simplify the equation. Once combined, you can solve for \(x\) straightforwardly.

Solving Equations

Solving equations might seem daunting, but it's a logical process. This involves isolating the variable, which in this case is \(x\), to one side of the equation.

Here's a step-by-step way to solve them:

• First, we need to move all terms with the variable \(x\) to one side. In our exercise, adding \(6.84x\) to both sides is necessary.
• Next, simplify the terms. In this exercise, \(4.65x + 6.84x = 11.49x\).
• The same goes for constant terms, moving them to the opposite side results in \(13.57 + 4.79 = 18.36\).
• Finally, solve for \(x\) by isolating it, which involves dividing both sides by the coefficient (the number before \(x\)). So, \(x = 18.36 / 11.49\).

The key is systematically tackling each part of the equation, ensuring that whatever operation is done to one side, is similarly performed on the other. This keeps the equation balanced.

Rounding Numbers

Rounding numbers can simplify results, making them easier to work with in real-world situations. In many math problems, like the one provided, you are asked to round the result to a specified decimal place.

For rounding to the nearest hundredth:

• Look at the third decimal place. If the number is 5 or greater, round up the second decimal place by one.
• If it is less than 5, simply retain the second decimal place as it is.

For example, if your calculated \(x\) value was 1.597, you would round it up to 1.60 since the third digit (7) is greater than 5.

Rounding is essential not just in math exercises but in daily calculations where precision is critical. It ensures consistency and follows common arithmetic conventions. These rules help you to make accurate judgment calls in both math and real-world applications.