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Chapter 3: Problem 32

Solve and check each equation. $$30=-\frac{x}{2}$$

Short Answer

Expert verified
We are given the equation \(30 = -\frac{x}{2}\). To solve for x, we first multiply both sides by 2, giving us \(60 = -x\). Then, we multiply both sides by -1, yielding \(-60 = x\). We check our solution by plugging x = -60 back into the original equation, which simplifies to \(30 = 30\). Therefore, the solution is \(x = -60\).

Step by step solution

01

Eliminate the fraction

To eliminate the fraction, we multiply both sides of the equation by the denominator, which in this case is 2. This gives us the equation: \(2(30) = 2(-\frac{x}{2})\)
02

Simplify the equation

We now simplify the equation by carrying out the multiplications. On the left side, we get 2 times 30, which is 60. On the right side, the 2's cancel out, leaving just -x: \(60 = -x\)
03

Solve for x

Currently, the equation says \(60 = -x\). To solve for x, we need to get x by itself on one side of the equation. This can be done by multiplying both sides of the equation by -1. This gives us: \(-1(60) = -1(-x)\) This simplifies to: \(-60 = x\)
04

Check the solution

Now we need to check if our solution, x = -60, is correct. To do this, we plug the value of x back into the original equation and verify if both sides of the equation are equal: \(30 = - \frac{-60}{2}\) This simplifies to: \(30 = 30\) Since both sides of the equation are equal, our solution is correct: \(x = -60\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eliminating Fractions
When solving linear equations that contain fractions, a helpful strategy is to eliminate the fractions right away. This simplifies the equation, making it easier to solve. Let's break it down with a simple tip:
  • Identify the fraction's denominator. In our exercise, the denominator is 2 in the equation \(-\frac{x}{2}\).
  • Multiply every term in the equation by this denominator. This operation cancels the fraction, transforming it into a whole number or a new expression.
Multiplying both sides of the equation by 2, as in the exercise, clears the fraction resulting in \(60 = -x\). An additional benefit of this method is that it maintains the equation's equality since we're applying the same operation to both sides.
Multiplying Equations
Multiplying both sides of an equation by a number is a fundamental concept in algebra that helps us simplify and solve equations. This action keeps the balance of the equation intact, which is crucial for finding the correct solution.
  • Always multiply every term on both sides鈥攏ot just the one with the variable or the constant.
  • Maintain the balance by performing the multiplication uniformly. This consistency ensures that the equation's equality holds after transforming it.
In the exercise, multiplying each side by 2 eliminated the fraction \(-\frac{x}{2}\), effectively offering a simpler equation \(60 = -x\) that can be solved.
Checking Solutions
After solving an equation, it's a smart practice to check your solution. This ensures everything is correct and gives confidence in your answer. Here鈥檚 how you can test your solution elegantly:
  • Substitute the solution back into the original equation.
  • Calculate both sides of the equation to see if they equal.
In our example, by plugging \(x = -60\) back into the original equation \(30 = -\frac{x}{2}\), we perform the calculation and confirm that \(30 = 30\). This verification step confirms the solution is accurate, ensuring no mistakes were made in the simplification and solving process.
Simplification in Algebra
Simplification is the process of making equations easier to work with by reducing them to their simplest form. It involves combining like terms and executing arithmetic operations across the equation.
  • Always perform one operation at a time to maintain clarity.
  • Simplifying should gradually make the equation more manageable.
In the context of our exercise, simplifying allowed us to reach the equation \(60 = -x\). By applying -1 multiplication on both sides, we easily isolated x, resulting in \(-60 = x\). Simplification not only aids in solving equations but also improves understanding of algebraic manipulations, making this skill invaluable for anyone tackling linear equations.

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