Problem 46 - College Algebra

Question

Solve for \(h: A=\frac{1}{2} b h\)

Step-by-Step Solution

Answer

The solution for \(h\) is \(h = \frac{2A}{b}\).

Step 1: Understand the Formula Given

You have an equation that defines the area \(A\) of a triangle: \(A = \frac{1}{2}bh\). Our goal is to solve for \(h\), which represents the height of the triangle.

Step 2: Isolate the Term with \(h\)

To begin solving for \(h\), we must first eliminate the fraction by multiplying both sides by 2 to get rid of the \(\frac{1}{2}\). This results in \(2A = bh\).

Step 3: Solve for \(h\)

Now, you need \(h\) by itself. To isolate \(h\), divide both sides of the equation by \(b\). This gives us \(h = \frac{2A}{b}\). Thus, \(h\) is expressed in terms of \(A\) and \(b\).

Key Concepts

Solving Equations

Solving equations involves finding the value of an unknown variable that satisfies the equation. This process is like solving a puzzle, where you use logical steps to reveal the unknown piece.

In our original exercise, we aimed to solve for the height 'h' in the triangle area formula: \( A = \frac{1}{2} b h \). The equation represents the area of a triangle where 'A' is the area, 'b' is the base, and 'h' is the height. To find 'h', we follow logical steps to manipulate the equation, allowing us to solve for this specific variable.

The key steps in solving equations involve understanding the equation first, then applying operations to isolate the variable you want to solve for. Through these systematic operations, the equation is restructured so that the unknown stands alone on one side of the equation.​

Isolating Variables

Isolating variables means manipulating an equation so that a specific variable appears alone on one side of the equation. This process typically involves using inverse operations, which are operations that reverse each other, like addition and subtraction or multiplication and division.

In our task, the goal was to solve for 'h' from the equation \( A = \frac{1}{2} b h \). To isolate 'h', we first eliminated the fraction by multiplying both sides by 2, converting the equation to \( 2A = bh \). This step cleared out the fraction and simplified the equation.

Next, we moved on to divide both sides by 'b', which finally left us with \( h = \frac{2A}{b} \). This outcome shows that 'h' is now isolated and can be calculated using known values of 'A' and 'b'.

Isolating variables is a fundamental skill in algebra, helping you expand your problem-solving toolkit.

Algebraic Manipulation

Algebraic manipulation involves the use of mathematical techniques to rearrange expressions and equations. It helps solve problems more easily and understand the relationships between different variables.

In the original exercise, algebraic manipulation was used to transform the given equation from \( A = \frac{1}{2} b h \) to \( h = \frac{2A}{b} \). This involves applying certain operations, like multiplying, dividing, adding, or subtracting both sides of an equation with the same value, to maintain the equality and achieve a desired form.

Key rules during algebraic manipulation include:

• Doing the same operation to both sides of an equation ensures balance is maintained.
• Use inverse operations to undo addition or multiplication within an equation.
• Reorganize terms to group like terms together if necessary.

Through efficient algebraic manipulation, solving complex problems becomes more systematic and manageable, allowing you to focus on understanding relationships instead of getting bogged down by calculation details.