Problem 76 - Technical Mathematics with Calculus

Question

Solve for \(x.\) Simple Literal Equations. $$c-b x=a-3 b$$

Step-by-Step Solution

Answer

The solution for x is given by dividing a-2b-c by b, hence x=(a-2b-c)/b.

Step 1: Isolate the term with the variable

To solve for the variable x, first isolate the term with x. You can do this by adding bx to both sides of the equation.

Step 2: Simplify both sides of the equation

After adding bx to both sides, the equation becomes c=a-2b+bx. Now, rearrange the terms to group the variable on one side: bx=a-2b-c.

Step 3: Solve for the variable

To find the value of x, divide both sides of bx=a-2b-c by b, assuming b is not equal to zero. This isolates x and gives you the solution for the variable.

Key Concepts

Isolate the Variable

One of the fundamental techniques in algebra is learning to isolate the variable. This means rearranging an equation so that one variable stands alone on one side of the equals sign, paving the way for solving the equation.

For instance, consider the equation \(c-bx=a-3b\). To isolate \(x\), you need to remove all other terms from its side. Adding \(bx\) to both sides leads to \(c=a-2b+bx\). Notice how \(x\) is now accompanied only by terms that include \(b\) on one side. That's the 'isolation' part—it's about giving the variable its own space. A key reminder is to perform the same operation on both sides to maintain the balance of the equation. Isolating the variable is crucial and forms the backbone of solving any algebraic equation.

Simplify Equations

Once you have initiated the isolation of the variable, the next step is to simplify the equation. This process often involves combining like terms and rearranging the equation into a more manageable form.

Using the previous example, after moving \(bx\) over to join its kind on the opposite side of the equation, our equation looks like this: \(c=a-2b+bx\). To simplify, you want to get \(x\) by itself by getting rid of the coefficients and constants associated with it. This is readily achieved by reversing the operations that are being performed on \(x\). Since it is being multiplied by \(b\), you'd ultimately divide by \(b\) to free \(x\) from this multiplication, assuming of course \(b\) is not zero. Simplifying ensures that the solution to the variable is clearly visible and leaves no room for error.

Literal Equations Step by Step

Taking a structured, step-by-step approach to solving literal equations can minimize mistakes and clarify the process. This methodical way of tackling equations helps students gain confidence and a deeper understanding of algebra.

Here is a breakdown using our given equation \(c-bx=a-3b\):

• Step 1: Add \(bx\) to both sides to group the variable term on one side, yielding \(c=a-2b+bx\).
• Step 2: Rearrange the terms to highlight the variable, resulting in \(bx=a-2b-c\).
• Step 3: Divide each side by \(b\) (don't forget to check that \(b\) isn't zero) to isolate \(x\) completely, which gives \(x=\frac{a-2b-c}{b}\).

Following these steps in sequence ensures a clear pathway from the original equation to a solution. Tackling equations in this manner is about understanding each move's purpose and ensuring that the final position of every term makes sense for isolating the desired variable.