Solve for g -6g+5=4g-2-3g

Solution:

Step 1:

Reposition the equation so that $g$ is on the left by rewriting $4g - 2 - 3g = -6g + 5$ as $-6g + 5 = 4g - 2 - 3g$.

Step 2:

Combine like terms on the right side: $4g - 3g$ simplifies to $g - 2 = -6g + 5$.

Step 3:

Consolidate all $g$ terms on one side of the equation.

Step 3.1:

Add $6g$ to each side to get $g + 6g - 2 = 5$.

Step 3.2:

Combine the $g$ terms to form $7g - 2 = 5$.

Step 4:

Isolate the $g$ term by moving constants to the opposite side.

Step 4.1:

Add $2$ to both sides to get $7g = 5 + 2$.

Step 4.2:

Combine the constants to simplify to $7g = 7$.

Step 5:

Solve for $g$ by dividing both sides of the equation by $7$.

Step 5.1:

Divide the equation $7g = 7$ by $7$ to get $\frac{7g}{7} = \frac{7}{7}$.

Step 5.2:

Reduce the left side by eliminating the common factor of $7$.

Step 5.2.1:

Cancel out the $7$s to simplify to $\frac{\cancel{7}g}{\cancel{7}} = \frac{7}{7}$.

Step 5.2.1.1:

Recognize that dividing $g$ by $1$ leaves $g$ unchanged, so $g = \frac{7}{7}$.

Step 5.3:

Simplify the right side by dividing $7$ by $7$ to find that $g = 1$.

Knowledge Notes:

The problem-solving process involves several key algebraic concepts:

1. Rearranging Equations: The ability to move terms from one side of an equation to the other while maintaining equality is fundamental. This often involves adding or subtracting the same term on both sides of the equation.

2. Combining Like Terms: When solving equations, it's important to combine terms that have the same variable raised to the same power. For example, $4g - 3g$ combines to $g$.

3. Isolating the Variable: The goal in solving an equation is to isolate the variable on one side to determine its value. This often requires moving constants (numbers without variables) to the opposite side of the equation.

4. Simplification: This step involves reducing fractions or canceling out common factors to simplify expressions.

5. Division Property of Equality: When both sides of an equation are divided by the same nonzero number, the two sides remain equal. This property is used to solve for the variable once it is isolated.

6. Latex Formatting: Mathematical expressions are formatted using Latex to clearly present equations and solutions. For example, $\frac{7g}{7} = \frac{7}{7}$ is a Latex-rendered equation.

Understanding and applying these concepts are essential for solving algebraic equations effectively.