Problem 33 - Prealgebra

Question

Solve the equation. \(-4.06 x-7.38=4.94 x\)

Step-by-Step Solution

Answer

\(x = -0.82\)

Step 1: Move like terms to one side

To solve for \(x\), we need to get all terms involving \(x\) on one side of the equation. Let's subtract \(4.94x\) from both sides to achieve that. The equation becomes: \[-4.06x - 4.94x - 7.38 = 0\]

Step 2: Simplify the equation

Now, combine the \(x\) terms on the left side of the equation: \(-4.06x - 4.94x = -9x\). So the equation now reads: \[-9x - 7.38 = 0\]

Step 3: Isolate the term with \(x\)

Add \(7.38\) to both sides to isolate the term with \(x\) on the left-hand side:\[-9x = 7.38\]

Step 4: Solve for \(x\)

Divide both sides by \(-9\) to solve for \(x\):\[x = \frac{7.38}{-9}\] Calculating this gives:\[x = -0.82\]

Step 5: Verify the solution

Since the exercise involves basic arithmetic, verifying the result can offer additional assurance. Substitute \(x = -0.82\) back into the original equation to verify:\(-4.06(-0.82) - 7.38 = 4.94(-0.82)\)Confirm that both sides give the same value when calculated.

Key Concepts

Equation Solving

Equation solving involves finding the value of the variable that makes the equation true. In the case of this exercise, our goal was to figure out what value of \(x\) satisfies the equation \(-4.06 x - 7.38 = 4.94 x\). The process starts by simplifying the equation, moving terms around to get all \(x\) terms on one side. By doing this, it becomes easier to isolate \(x\) and subsequently find its value. This method is useful for solving linear equations, i.e., equations where the variable is raised only to the first power. By following systematic steps, you can solve a wide range of equations effectively.

Basic Arithmetic

To solve equations, a firm grasp of basic arithmetic operations—addition, subtraction, multiplication, and division—is essential. For instance, in this exercise, subtracting and combining like terms is critical for simplifying the equation. This involves understanding negative numbers and managing operations with decimals. When we calculated the terms,

• we subtracted \(4.94x\) from both sides,
• combined the \(-4.06x\) and \(-4.94x\) to yield \(-9x\).

Later, basic arithmetic helped isolate and solve for \(x\) by dividing both sides by \(-9\). This reliance on arithmetic showcases how foundational these skills are in equation-solving, as they provide the groundwork for manipulating numerical expressions effectively.

Isolating Variables

Isolating the variable is crucial when solving equations. Here, that means getting \(x\) alone on one side of the equation. Initially, both sides had terms with \(x\), which led us to subtract \(4.94x\) from both sides to consolidate the \(x\) terms. This produced a cleaner expression: \(-9x - 7.38 = 0\).
Further, by adding \(7.38\) to both sides, the equation simplified to \(-9x = 7.38\). Finally, dividing by \(-9\) gave us the value of \(x\). Isolating variables makes solving equations more straightforward. It involves using inverse operations like addition to counter subtraction or division to counter multiplication, enabling us to uncover the variable's value.

Verification of Solutions

Once a solution is found, it's important to verify it. Verification confirms the accuracy of your solution. In this exercise, substituting \(x = -0.82\) back into the original equation ensures both sides equal when calculated. This computation involves checking if:

• \(-4.06(-0.82) - 7.38\) equals \(4.94(-0.82)\).

By calculating both sides, we affirm they match, confirming \(x = -0.82\) is indeed correct. Verification can involve substituting results back into the initial equation or double-checking arithmetic operations. This step is not only about accuracy but also about boosting confidence in mathematical skills. If verified correctly, one can be assured their understanding and process were spot on.