Problem 24 - Introductory Algebra for College Students

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2} h(a+b)\) for \(a\)

Step-by-Step Solution

Answer

The side \(a\) in the trapezoid area formula can be isolated as \(a=\frac{2A}{h}-b\).

Step 1: Cancel the Fraction

First, every term in the equation should be multiplied by 2 to remove the fraction. This changes the formula to \(2A=h(a+b)\).

Step 2: Divide by h

Divide both sides of the equation by \(h\) to isolate the terms involving \(a\). This results in \(\frac{2A}{h}=a+b\).

Step 3: Solve For a

Finally, subtract \(b\) from both sides of the equation to isolate \(a\). This gives the final equation as \(a=\frac{2A}{h}-b\).

Key Concepts

Formula Manipulation

Formula manipulation is all about changing how a formula looks without changing what it means. It's like rearranging the furniture in your room—you still have the same pieces but you might have a better setup for what you need. In algebra, we manipulate formulas to solve for a specific variable or to make them easier to use.Consider this example: we have the formula \(A = \frac{1}{2} h(a+b)\), and we need to solve it for \(a\). The first step is removing the fraction by multiplying every term by 2. This gets rid of fractions, which are tricky to work with, and transforms our equation into \(2A = h(a+b)\). Now the equation is more straightforward, and we can continue to isolate \(a\). This kind of manipulation is the backbone of solving equations because it simplifies our tasks and helps us to focus.

Solving Equations

Solving equations is like solving a mystery—every clue you uncover gets you closer to the answer. Once we have the equation \(2A = h(a+b)\), we aim to isolate the variable we're interested in, which is \(a\). This is done by rearranging the equation step by step.

• Divide both sides by \(h\) to isolate the term that contains \(a\).
• This gives us \(\frac{2A}{h} = a + b\).

Now you see the path to \(a\)—all that's left is subtracting \(b\) from each side:

• \(a = \frac{2A}{h} - b\)

By following these steps in order, we've effectively uncovered the value of \(a\). It's important to work methodically to ensure no terms are lost or misplaced in the process.

Geometry Formulas

Geometry formulas help us understand the world of shapes and spaces. The initial formula \(A = \frac{1}{2} h(a+b)\) is derived from geometry and is known for calculating the area of a trapezoid. Recognizing this formula is helpful, as it can give context to what we're solving for.Each piece of the formula has a specific meaning:

• \(A\) is the area you're trying to find.
• \(h\) represents the height of the trapezoid.
• \(a\) and \(b\) are the lengths of the two parallel sides (bases) of the trapezoid.

By manipulating this formula, we not only solve algebraic equations but also bridge into geometrical insights. Understanding and identifying standard geometry formulas can greatly enhance your problem-solving skills across different areas of math.