Problem 44 - Algebra 1

Question

Solve the equation. Round the result to two decimal places. $$17.87-2.87 x=1.87-4.92 x$$

Step-by-Step Solution

Answer

The value of 'x' rounded to two decimal places is approximately 7.80

Step 1: Collect like terms

Move all terms containing the variable 'x' to one side and numerical terms to the other. We get: \(2.87x - 4.92x = 1.87 - 17.87 \)

Step 2: Simplify each side

Combine like terms on each side to simplify. For the left side, when you subtract the coefficients, you get \(-2.05x\). And for the right side, when you subtract 17.87 from 1.87, you get \(-16.00\) . So the equation becomes: \(-2.05x = -16.00\)

Step 3: Isolate for 'x'

To isolate 'x', divide both sides by the coefficient of 'x', that is -2.05. So: \(x = -16.00 / -2.05 \)

Step 4: Compute the value of 'x'

Perform the division to get the value of 'x', rounding to two decimal places. The result of the division is approximately \(7.80\)

Key Concepts

Collect Like Terms

Understanding how to \textbf{collect like terms} is essential in solving linear equations. Imagine you have a pile of apples and oranges mixed together. To know how many of each fruit you have, you separate them into two piles: one for apples and one for oranges. The same principle applies to algebraic expressions.

Let's look at the expression on one side of the given equation: \(17.87 - 2.87x\). Here, 'like terms' refer to the terms that have the same variable raised to the same power or constants (terms without variables). In this case, to begin simplifying the equation, we need to get all the 'x' terms on one side and the constants on the other. This organization makes the equation clearer and sets the stage for combining those terms effectively.

Combining Like Terms

The next crucial step is \textbf{combining like terms}. This technique involves simplifying algebraic expressions by adding or subtracting like terms. To successfully combine like terms, they must have exactly the same variable component.

For instance, in the given problem after moving the terms, we ended up with an equation that looked like this: \(2.87x - 4.92x = 1.87 - 17.87\). On each side of the equation, we combine like terms to make our equation more manageable. We subtract the coefficients of 'x' terms (since they are like terms) to get \(-2.05x\), and we subtract the constants to get \(-16.00\). This simplification is vital as it brings us closer to finding the value of 'x'.

Isolating Variables

The final step to solving a linear equation is \textbf{isolating the variable}. This means manipulating the equation so that the variable you're solving for, in this case 'x', stands alone on one side of the equation. This action gives you the solution to the equation.

From the combined equation \(-2.05x = -16.00\), we want to get 'x' by itself. To do this, we divide both sides by the coefficient of 'x', which is -2.05. By performing this division, we isolate 'x' and solve for its numerical value. Remember to maintain the balance of the equation: whatever operation you do to one side, you must do to the other. Ultimately, we find that \(x = 7.80\), rounded to two decimal places, is the solution.