Solve the following eqations.Check your result in each case
`2(X-2)+3(4X-1)=0`

Answer

To solve the equation \(2(X-2)+3(4X-1)=0\), we will follow these steps:

### Step 1: Expand the equation
Start by distributing the constants outside the parentheses:

\[ 2(X-2) + 3(4X-1) = 0 \]

Expanding this gives:

\[ 2X - 4 + 12X - 3 = 0 \]

### Step 2: Combine like terms
Now, combine the like terms (the terms with \(X\) and the constant terms):

\[ (2X + 12X) + (-4 - 3) = 0 \]

This simplifies to:

\[ 14X - 7 = 0 \]

### Step 3: Isolate the variable
Next, we want to isolate \(X\). Start by adding 7 to both sides:

\[ 14X = 7 \]

Now, divide both sides by 14:

\[ X = \frac{7}{14} \]

Simplifying this gives:

\[ X = \frac{1}{2} \]

### Step 4: Check the result
To check our solution, substitute \(X = \frac{1}{2}\) back into the original equation:

\[ 2\left(\frac{1}{2}-2\right) + 3\left(4\left(\frac{1}{2}\right)-1\right) = 0 \]

Calculating inside the parentheses first:

\[ 2\left(\frac{1}{2} - 2\right) = 2\left(\frac{1}{2} - \frac{4}{2}\right) = 2\left(-\frac{3}{2}\right) = -3 \]

And for the second part:

\[ 3\left(4\left(\frac{1}{2}\right) - 1\right) = 3\left(2 - 1\right) = 3 \times 1 = 3 \]

Now, combine the results:

\[ -3 + 3 = 0 \]

Since both sides of the equation are equal, our solution is verified.

### Final Solution:
\[ X = \frac{1}{2} \]

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