Problem 46 - Algebra 1

Question

Solve the equation. $$-\frac{1}{2} b=-|-8|$$

Step-by-Step Solution

Answer

The solution to the equation is b = -16.

Step 1: Understand the Absolute Value

In an equation, the absolute value of a number (in this case -8) is the positive version of that number. So the absolute value of -8 is 8.

Step 2: Rewrite the Equation

Substitute the absolute value of -8 into the equation. So the equation becomes -\(\frac{1}{2}\) b = 8.

Step 3: Solve for the Variable b

To solve for b, you should isolate b by multiplying both sides by the opposite of denominator, which is -2. So b = 8 x -2 = -16.

Key Concepts

Absolute Value

Understanding the absolute value is essential when dealing with equations that involve it. The absolute value of a number refers to its distance from zero on the number line, regardless of direction. In simpler terms:

• It is always non-negative.
• The absolute value of -8 is 8, because the distance from -8 to 0 is 8 units.
• Similarly, the absolute value of any positive number is the number itself, because that is already its distance to zero.

This process is crucial in solving equations, since incorrectly interpreting absolute values can lead to wrong answers. Always replace the absolute value of a number with its non-negative counterpart in equations.

Isolate the Variable

To solve an equation for a specific variable, such as 'b' in this problem, you need to isolate it on one side of the equation. Isolating the variable means rearranging the equation so that the variable you are solving for stands alone. When you have an equation like -\(\frac{1}{2}\)b = 8, follow these steps:

• First, analyze the equation to identify the operations performed on the variable.
• In this case, 'b' is being multiplied by -\(\frac{1}{2}\).
• To isolate 'b,' you must get rid of this fraction by performing the opposite operation – which is multiplication by its reciprocal.

After carrying out these steps correctly, you'll have 'b' isolated, making it easy to determine its value.

Multiplication Rule

The multiplication rule is instrumental when you need to isolate a variable, especially in dealing with fractions. Here's how it applies in this scenario:

• Begin with the equation -\(\frac{1}{2}\)b = 8.
• You want to eliminate the -\(\frac{1}{2}\) coefficient of 'b'.
• Multiply both sides of the equation by the reciprocal of -\(\frac{1}{2}\), which is -2.

This step is crucial because multiplying by the reciprocal effectively cancels out the fraction. Therefore,

• \(-\frac{1}{2} \cdot -2 = 1\), leaving 'b' on one side.
• Don't forget to apply the same multiplication to the other side of the equation: 8 \(\times\) -2 = -16.

This will provide you with the final value of 'b': -16. Always remember to apply multiplication or any operation equally to both sides of the equation to maintain balance.