Problem 35 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. $$ 39.21 x+2.65=-31.68+42.03 x $$

Step-by-Step Solution

Answer

After analyzing and executing the steps, the solution to the algebraic equation is \(x = 0.94\).

Step 1: Rearrange the equation

Firstly, collect all terms with \(x\) on one side of the equality sign and the constants on the other side. In order to do this, subtract \(39.21x\) from both sides of the equation and add \(31.68\) to both sides of the equation, resulting in \(42.03x - 39.21x = -31.68 + 31.68 + 2.65\).

Step 2: Solve for \(x\)

Now simplify. You get \(2.82x = 2.65\). Dividing both sides of the equation by \(2.82\) to isolate \(x\) you get \(x = 2.65/2.82\).

Step 3: Round the result

Evaluate \(2.65/2.82\) and round the answer to the nearest hundredth as instructed. This gives \(x = 0.94\).

Key Concepts

Linear Equations

Linear equations are foundational in algebra and form the basis for more complex mathematical concepts. A linear equation represents a straight line when graphed on a coordinate plane and is composed of variables, constants, and operations such as addition and subtraction. In its simplest form, a linear equation with one variable looks like \(ax + b = 0\)), where \(a\)) and \(b\)) are constants, and \(x\)) is the variable we want to solve for.

In the exercise provided, the equation \(39.21x + 2.65 = -31.68 + 42.03x\)) is a linear equation with a single variable, \(x\)). The goal is to find the value of \(x\)) that satisfies the equation. To solve linear equations, you typically simplify and manipulate the equation using basic arithmetic until you isolate the variable on one side of the equal sign. This process involves eliminating terms and combining like terms sensibly to make the equation easier to solve, often leading to a solution where \(x = \text{some number}\)).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. These expressions can represent quantities that vary, which we call variables. For example, in the given equation, \(39.21x + 2.65\)) and \(42.03x\)) are algebraic expressions that denote two quantities depending on the value of \(x\)).

Unlike equations, algebraic expressions do not have an equal sign. They are manipulated through algebraic laws and properties such as the distributive property, the commutative property of addition and multiplication, and the associative property. Understanding algebraic expressions is essential in the process of solving linear equations, as it allows you to rearrange and simplify these expressions effectively.

Equation Solving Step by Step

Solving equations step by step is a systematic method used to find the solution to equations, including linear ones. Each step is designed to gradually simplify the equation, making it more straightforward to solve.

In the exercise example, step 1 focuses on restructuring the equation to gather like terms together. This means moving the terms containing \(x\)) to one side and the constant numbers to the other side. Step 2 then simplifies the new equation by combining the like terms, which subsequently isolates the variable term. Finally, in step 3, once the variable is isolated, simple arithmetic gives you the value of the variable. This step-by-step method ensures a thorough and understandable approach to solving equations, catering well to beginners in algebra.

When executing these steps, it's crucial to perform the same operation on both sides of the equation to maintain equality. Additionally, it's important to understand the properties of equality and arithmetic operations, as these will always guide you toward the correct next step in the process.