Problem 9 - Intermediate Algebra

Question

Complete the solution. Solve \(t=a d+b c\) for \(c\) $$ \begin{aligned} &t-\quad=a d+b c-\\\ &t-a d=\\\ &t-a d\\\ &\frac{t-a d}{b}=\\\ &c= \end{aligned} $$

Step-by-Step Solution

Answer

\( c = \frac{t - a d}{b} \)

Step 1: Write the Equation

We start with the equation given: \( t = a d + b c \). Our goal is to solve for \( c \).

Step 2: Isolate the term with c

Subtract \( a d \) from both sides of the equation to isolate the term \( b c \) on the right side: \( t - a d = b c \).

Step 3: Solve for c

To isolate \( c \), divide both sides of the equation by \( b \). The equation becomes: \( c = \frac{t - a d}{b} \).

Key Concepts

Isolating Variables

In solving linear equations, isolating variables is often a crucial step. It involves rearranging the equation to get the variable of interest alone on one side of the equation. This makes it easier to solve for that variable. For example, consider the equation \( t = a d + b c \). We need to solve for \( c \). The first step is to isolate the term \( b c \) on one side of the equation. By subtracting \( a d \) from both sides, we effectively move \( a d \) to the opposite side:

• Original equation: \( t = a d + b c \)
• Subtract \( a d \) from both sides: \( t - a d = b c \)

By isolating \( b c \), we have set up the equation perfectly for the next step, where we'll focus on \( c \). Remember, isolating variables helps simplify equations to make them more manageable and easier to solve.

Equation Manipulation

Equation manipulation is all about using algebraic operations to transform and simplify equations. It involves applying mathematical operations like addition, subtraction, multiplication, and division to both sides of an equation to shape it into a form that can be more easily solved. In our example, after isolating \( b c \), we manipulate the equation further to solve for \( c \).To solve for \( c \), we need to divide both sides by the coefficient \( b \), which accompanies the variable \( c \). This operation will effectively "cancel out" \( b \) on the side of the equation with \( c \):

• Equation before manipulation: \( t - a d = b c \)
• Divide each side by \( b \): \( \frac{t - a d}{b} = c \)

This manipulation leaves \( c \) isolated on one side of the equation, which completes our solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding how to work with these is key to solving equations. In the equation \( t = a d + b c \), each term—\( a d \), \( b c \), and \( t \)—is an algebraic expression. These expressions can contain coefficients (like \( a \) and \( b \)), which are numbers that modify variables.To work with algebraic expressions, one must be comfortable with rearranging them by:

• Identifying like terms. For example, in \( a d + b c \), neither term can be combined directly because they have different variables.
• Using operations such as addition, subtraction, multiplication, and division. These help in moving parts of the expression around to solve for the variable we are interested in.

By becoming familiar with algebraic expressions, you develop the ability to manipulate and solve equations efficiently, as we did by isolating \( c \) in the original equation.