Problem 30 - Prealgebra

Question

Solve equation. Check your solution. \(0.4 x=2 x+1.2\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -0.75\).

Step 1: Set the equation

We start with the equation given in the problem: \[0.4x = 2x + 1.2\]

Step 2: Move all terms involving x to one side

Subtract \(2x\) from both sides of the equation to keep all terms involving \(x\) on one side:\[0.4x - 2x = 1.2\]

Step 3: Simplify the equation

Combine like terms on the left side of the equation:\[-1.6x = 1.2\]

Step 4: Solve for x

Divide both sides of the equation by \(-1.6\) to solve for \(x\):\[x = \frac{1.2}{-1.6}\]

Step 5: Simplify the fraction

Simplify the fraction \(\frac{1.2}{-1.6}\):Convert to decimals or simplify: \[x = -\frac{3}{4}\] or \(x = -0.75\).

Step 6: Check the solution

Substitute \(x = -0.75\) back into the original equation to ensure it satisfies the equation:\[0.4(-0.75) = 2(-0.75) + 1.2\]Calculate both sides:\[-0.3 = -1.5 + 1.2\]\[-0.3 = -0.3\]Since both sides are equal, the solution is correct.

Key Concepts

Prealgebra

Prealgebra is an essential foundational branch of mathematics focused on establishing the basic concepts of algebra. This stage introduces students to understanding numbers and their operations, which sets the stage for more complex mathematical ideas. Prealgebra covers the manipulation of arithmetic operations involving whole numbers, fractions, decimals, and variables. A key focus in prealgebra is understanding variables, represented by letters, usually like "x" or "y." These variables stand for unknowns that you try to solve in an equation. Prealgebra begins with solving simple equations and inequalities, where you learn techniques to find what the variable represents. Such exercises lay the groundwork for future algebraic learning. In essence, prealgebra acts as the gateway to understanding broader math concepts making it crucial for young students.

Checking Solutions

One of the most critical steps in solving equations is checking your solution. This means ensuring that the value you find for the variable truly satisfies the original equation. Checking the solution involves:

• Substituting your solution back into the original equation.
• Performing the arithmetic operations as instructed by the equation.
• Comparing both sides of the equation to see if they're equal.

For example, if your original equation is \(0.4x = 2x + 1.2\) and your solution is \(x = -0.75\), you should replace every "x" in the original equation with \(-0.75\). Calculating both sides helps determine the equality, confirming whether your solution is correct. If both sides match, you've likely found a correct answer. If not, revisiting your steps is necessary  perhaps re-examining calculations or how you simplified complex parts of the equation. Checking solutions confidently closes the loop in problem-solving, ensuring accuracy and understanding.

Equation Simplification

Simplifying an equation is a core skill in algebra that involves making the equation as straightforward as possible. This means reducing the equation to its simplest form without changing its solutions.The process includes:

• Combining like terms where applicable. For instance, in \(0.4x - 2x\), combining these gives \(-1.6x\).
• Performing operations like addition, subtraction, multiplication, or division across the equation.
• Ensuring to mirror operations across both sides to maintain equality.

The goal is to isolate the variable, simplifying the path to finding its value. For example, in the exercise provided, when we subtract \(2x\) from both sides, we simplify the terms involving "x" on one side, leading us from \(0.4x = 2x + 1.2\) to \(-1.6x = 1.2\).Simplifying is not only about finding solutions faster but also creating a more intuitive understanding of the equation's structure. Through simplification, the complexities of equations are broken down and made digestible, paving the way for clearer, more precise problem-solving.