Solve for x 6(2x-1)-(x-4)=-2(3x+1)

Solution:

Step 1: Simplify the left-hand side of the equation $6(2x-1)-(x-4)$.

Step 1.1: Break down each term.

Step 1.1.1: Distribute $6$ over $(2x-1)$ and subtract $(x-4)$ to get $6(2x)+6(-1)-(x-4)=-2(3x+1)$.

Step 1.1.2: Compute $6\times 2$ to obtain $12x+6(-1)-(x-4)=-2(3x+1)$.

Step 1.1.3: Compute $6\times -1$ to get $12x-6-(x-4)=-2(3x+1)$.

Step 1.1.4: Distribute the negative sign over $(x-4)$ to find $12x-6-x+4=-2(3x+1)$.

Step 1.1.5: Simplify to $12x-6-x+4=-2(3x+1)$.

Step 1.2: Combine like terms.

Step 1.2.1: Combine $12x-x$ to get $11x-6+4=-2(3x+1)$.

Step 1.2.2: Add $-6$ and $4$ to simplify further to $11x-2=-2(3x+1)$.

Step 2: Simplify the right-hand side of the equation $-2(3x+1)$.

Step 2.1: Use the distributive property to expand $-2(3x+1)$.

Step 2.2: Perform the multiplication.

Step 2.2.1: Multiply $3$ by $-2$ to get $11x-2=-6x-2(1)$.

Step 2.2.2: Multiply $-2$ by $1$ to simplify to $11x-2=-6x-2$.

Step 3: Isolate terms with $x$ on one side.

Step 3.1: Add $6x$ to both sides to get $11x+6x-2=-2$.

Step 3.2: Combine $11x$ and $6x$ to find $17x-2=-2$.

Step 4: Move constant terms to the other side.

Step 4.1: Add $2$ to both sides to obtain $17x=-2+2$.

Step 4.2: Combine $-2$ and $2$ to find $17x=0$.

Step 5: Solve for $x$.

Step 5.1: Divide both sides by $17$ to get $\frac{17x}{17}=\frac{0}{17}$.

Step 5.2: Simplify the left side.

Step 5.2.1: Cancel out the common factor of $17$.

Step 5.2.1.1: Reduce to $\frac{\cancel{17}x}{\cancel{17}}=\frac{0}{17}$.

Step 5.2.1.2: Simplify to $x=\frac{0}{17}$.

Step 5.3: Simplify the right side to find $x=0$.

Knowledge Notes:

The problem-solving process involves several mathematical concepts and techniques:

1. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to expand expressions by multiplying each term inside the parentheses by the term outside.

2. Combining Like Terms: This refers to the process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power.

3. Isolating the Variable: This involves moving all terms containing the variable to one side of the equation and all constant terms to the other side to make it easier to solve for the variable.

4. Simple Arithmetic: Basic arithmetic operations such as addition, subtraction, multiplication, and division are used throughout the problem-solving process.

5. Equation Solving: The ultimate goal is to find the value of the variable that satisfies the equation. This often involves isolating the variable on one side of the equation.

6. Checking Solutions: After finding a potential solution, it's important to check that it satisfies the original equation to ensure that it is correct.