Problem 61 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. $$1.29 x=5.22 x+3.61$$

Step-by-Step Solution

Answer

After evaluating the expression in step 3, rounding the result to the nearest hundredth, x is approximately equal to \(0.92\).

Step 1: Rearrange the equation

Subtract \(5.22x\) from both sides of the equation to gather all terms containing the unknown x on one side. The equation becomes:\(-3.93x = 3.61\)

Step 2: Solve for x

Divide both sides of the equation by -3.93 to solve for x. The equation becomes: \(x = -3.61/-3.93\)

Step 3: Evaluate the expression

Evaluate the expression using a calculator to get the final answer for x, and round the result to the nearest hundredth.

Key Concepts

Equation Rearrangement

Rearranging an equation is often the first step when solving equations with variables. Let's take a closer look at what equation rearrangement involves. In our equation \(1.29x = 5.22x + 3.61\), our goal is to isolate the variable \(x\) on one side of the equation. This usually means moving all terms with \(x\) to one side and constants to the other. To do this, we use the principle of maintaining equality by performing the same operation on both sides of the equation. Here, we need to eliminate \(5.22x\) from the right side. To achieve that, subtract \(5.22x\) from both sides, resulting in \(-3.93x = 3.61\). By simplifying and rearranging equations in this manner, we position ourselves to the final solution more clearly. This rearrangement highlights the valuable ability to manipulate and control equation variables effectively.

Rounding Numbers

Rounding numbers is an essential skill in mathematics, especially when dealing with long decimal answers. Once we find the solution for \(x\) in our equation, it may contain several decimal places. To simplify, we round to a specified place value. In this case, we round to the nearest hundredth. When rounding, look at the digit immediately after the target decimal place. Here, if it is 5 or above, we round up, otherwise, we round down. For example, if we arrive at a number like 0.9173, the hundredth digit is 1, and the next digit is 7; thus, we round to 0.92. Understanding how to round numbers allows us to present answers that are easier to interpret, ensuring consistency in precision across various calculations.

Algebraic Expressions

In mathematics, algebraic expressions are combinations of variables, numbers, and operations. When solving equations, recognizing and handling these expressions is crucial. In our exercise \(1.29x = 5.22x + 3.61\), each term is part of a larger algebraic expression with the variable \(x\).Successful manipulation of algebraic expressions often involves adding, subtracting, and factoring these components. As we saw, we subtracted \(5.22x\) from both sides, simplifying the expression to \(-3.93x = 3.61\). Thus, balancing and simplifying expressions allows us to solve for the variable effectively.Skills in working with algebraic expressions enable one to change complex problems into simpler ones, paving the way for efficient and clear problem-solving.