Problem 74 - Elementary Algebra

Question

Solve for \(b: A=12 b h\)

Step-by-Step Solution

Answer

\(b = \frac{A}{12h}\)

Step 1: Understand the Equation

The given equation is the formula for the area of a triangle, where \(A\) represents the area, \(b\) is the base, and \(h\) is the height. We need to solve for \(b\). The equation to start with is \(A = 12b h\).

Step 2: Isolate the Variable

To solve for \(b\), we need to isolate \(b\) on one side of the equation. To do this, divide both sides of the equation by \(12h\). This gives us: \[\frac{A}{12h} = b\]

Step 3: Simplified Solution

After dividing both sides by \(12h\), we have isolated \(b\) on one side. The expression \(\frac{A}{12h}\) gives us the value of \(b\). Therefore, the solution is \(b = \frac{A}{12h}\).

Key Concepts

Area of a Triangle

The area of a triangle is one of the fundamental concepts in geometry. It's a measure of the space enclosed by the triangle's sides. The formula to calculate the area of a triangle is

• \( A = \frac{1}{2} \times b \times h \).

Here, \(A\) represents the area, \(b\) is the length of the base, and \(h\) is the height, which is perpendicular to the base.

To apply this formula effectively:

• Identify the base \(b\) and corresponding height \(h\) of the triangle.
• Substitute these values into the formula.
• Perform the multiplication to find the area.

The area helps in numerous applications, such as determining the amount of material needed for construction or calculating land sizes.

Understanding this simple calculation can deepen your knowledge of how triangles work within broader mathematical contexts like coordinate geometry or trigonometry.

Isolate the Variable

Isolating the variable is a systematic process in algebra where your aim is to get the variable by itself. This process allows you to solve equations more easily. In the context of the equation \( A = 12bh \), we want to solve for the base \(b\). Here’s how you can do it:

• Identify what you’re solving for, in this case, \(b\).
• Determine what operations are currently acting on \(b\). Here, \(b\) is being multiplied by \(12\) and \(h\).
• Use the inverse operation to eliminate these, which involves dividing both sides of the equation by \(12h\).

This step will effectively "free" \(b\), giving you:\[ b = \frac{A}{12h} \].

By isolating \(b\), you can now directly calculate it, given the values of \(A\) and \(h\). Isolating variables is crucial as it helps solve equations in a variety of areas like physics, economics, and engineering.

Equation Manipulation

Equation manipulation is a fundamental technique in mathematics that involves altering equations to make them more solvable or understandable. This can include rearranging or simplifying equations, just as we did while isolating the variable \(b\) in the exercise.

Manipulating equations involves a few key strategies:

• Rearranging terms to make sense of the given information.
• Performing valid mathematical operations on both sides of the equation (like addition, subtraction, multiplication, or division) to maintain equality.
• Applying properties of equality to systematically simplify or solve equations.

In our example, dividing both sides by \(12h\) maintained the balance of the equation, allowing us to isolate \(b\).

Equation manipulation is widely used across different mathematical disciplines. Mastering these techniques can turn seemingly complex problems into simple, manageable tasks. By practicing, you’ll enhance your problem-solving skills significantly.