Solve for g b=(tr-g)/r

Solution:

Step 1:

Express the equation in the form $\frac{tr - g}{r} = b$.

Step 2:

Remove the denominator by multiplying both sides by $r$: $(\frac{tr - g}{r}) \cdot r = b \cdot r$.

Step 3:

Eliminate the fraction on the left-hand side:

Step 3.1:

Consider $\frac{tr - g}{r} \cdot r$.

Step 3.1.1:

Remove the $r$ in the denominator and numerator: $\frac{tr - g}{\cancel{r}} \cdot \cancel{r} = br$.

Step 3.1.1.2:

The equation simplifies to $tr - g = br$.

Step 4:

Isolate $g$ on one side of the equation:

Step 4.1:

Subtract $rt$ from both sides: $-g = br - rt$.

Step 4.2:

Divide the entire equation by $-1$ to solve for $g$:

Step 4.2.1:

Apply the division: $-\frac{g}{-1} = \frac{br}{-1} - \frac{rt}{-1}$.

Step 4.2.2:

Simplify the equation:

Step 4.2.2.1:

Recognize that dividing negatives yields a positive: $\frac{g}{1} = -br + \frac{rt}{1}$.

Step 4.2.2.2:

Simplify further to get $g = -br + rt$.

Knowledge Notes:

1. Algebraic Manipulation: The process involves rearranging the equation and performing operations such as multiplication, division, addition, and subtraction to isolate the variable of interest.

2. Simplifying Fractions: When a fraction has the same factor in the numerator and the denominator, those factors can be canceled out.

3. Distributive Property: This property is used when multiplying a number by a sum or difference, allowing us to multiply the number by each term inside the brackets separately.

4. Negative Numbers: Dividing or multiplying two negative numbers results in a positive number.

5. Isolating Variables: The goal in algebraic equations is often to solve for one variable in terms of others. This involves moving terms from one side of the equation to the other and simplifying until the variable of interest is by itself.

6. Latex Formatting: Mathematical expressions are rendered using Latex to provide clear and professional-looking equations.