Question

Solve the equation. $$ \frac{1}{3} g-2=1 $$

Short answer

The value of 'g' is 9

Step-by-step solution

Set the equation

Start with the equation \( \frac{1}{3} g-2=1 \). Your main goal is to solve for 'g'.

Isolate terms with 'g'

Start isolating the terms containing 'g'. To do this, you need to eliminate -2 from the left side of the equation. You can do this by adding 2 to both sides of the equation. This will result in \( \frac{1}{3} g=1+2 \).

Simplify

Simplify the right side of the equation, i.e. perform the addition operation on the right side. This results in \( \frac{1}{3} g = 3 \).

Solve for 'g'

To find 'g', multiply each side of the equation by 3 to solve for 'g'. It becomes \( g = 3 * 3 \).

Evaluate the result

Finally, perform the multiplication operation on the right side of the equation to get the value of 'g'.

Key concepts

Isolating Variables

When we want to find the value of a variable in an equation, the first step is to isolate it. Isolating a variable means to get the variable by itself on one side of the equation, with no other variables or constants in its way. For example, in the equation \( \frac{1}{3} g - 2 = 1 \), to isolate the variable \( g \), we need to eliminate the -2. By adding 2 to both sides, we have \( \frac{1}{3} g = 1 + 2 \). This method is the crux of solving linear equations, as it simplifies the problem into a format where the variable in question stands alone and is easily identified.

Imagine you have several items on a balance scale, and your goal is to have the variable—let's call it 'X'—in the solo spotlight on one side of the scale. You'd strategically remove weights (constants and coefficients) from the side where 'X' is located by performing the same action on the other side to keep the scale balanced—that's what we're doing in algebra when we isolate variables.

Equation Simplification

Combining Like Terms

To simplify an equation, we combine like terms and perform arithmetic operations to reduce the equation to a simpler form without changing its equality. In the given exercise, after isolating the variable on one side, we simplify by adding numeric values together, changing \( \frac{1}{3} g = 1 + 2 \) to \( \frac{1}{3} g = 3 \). This step makes the equation easier to work with and brings us closer to finding the solution.

Clear Coefficients

It's also important to remove any coefficients (numbers multiplied by the variable), which is achieved by executing the inverse operation. If the variable is divided by a number, multiply both sides of the equation by that number. If it's multiplied by a number, divide both sides by that number. This results in an equation with just the variable equal to a number, cleanly presenting the solution.

Solving for Variables

Solving for a variable essentially means doing whatever mathematics necessary to find the value of that variable. Once we use equation simplification to get a variable by itself on one side of an equation, we perform operations like multiplication or division to both sides to solve for it. In our example, after simplifying the equation to \( \frac{1}{3} g = 3 \), we solve for \( g \) by multiplying both sides by 3 (the reciprocal of \( \frac{1}{3} \) that we had originally). This leaves us with \( g = 9 \), which is our final solution.

This method can be applied to any equation where the goal is to find the value of one variable. The beauty of algebra lies in these basic principles; no matter how complex an equation looks, breaking it down through isolation, simplification, and solving for the variable will often lead the way to the solution.