Solve the following equation.Check your result in each case
`0.5x+x/3=0.25x+7`

Answer

To solve the equation \(0.5x + \frac{x}{3} = 0.25x + 7\), we will follow these steps:

### Step 1: Find a common denominator
The denominators in the equation are 1 (for \(0.5x\) and \(0.25x\)) and 3 (for \(\frac{x}{3}\)). The least common multiple (LCM) of 1 and 3 is 3.

### Step 2: Rewrite the equation using the common denominator
Multiply each term by 3 to eliminate the fraction:
\[ 3(0.5x) + 3\left(\frac{x}{3}\right) = 3(0.25x) + 3(7) \]
This simplifies to:
\[ 1.5x + x = 0.75x + 21 \]

### Step 3: Combine like terms
On the left side, combine \(1.5x\) and \(x\):
\[ (1.5x + x) = 2.5x \]
So the equation now looks like:
\[ 2.5x = 0.75x + 21 \]

### Step 4: Move all \(x\) terms to one side
Subtract \(0.75x\) from both sides:
\[ 2.5x - 0.75x = 21 \]
This simplifies to:
\[ 1.75x = 21 \]

### Step 5: Solve for \(x\)
To isolate \(x\), divide both sides by \(1.75\):
\[ x = \frac{21}{1.75} \]

### Step 6: Convert the division to eliminate the decimal
To make the division easier, multiply the numerator and denominator by 100:
\[ x = \frac{21 \times 100}{1.75 \times 100} = \frac{2100}{175} \]

### Step 7: Simplify the fraction
Now simplify \(\frac{2100}{175}\):
\[ 2100 \div 25 = 84 \quad \text{and} \quad 175 \div 25 = 7 \]
So:
\[ x = \frac{84}{7} = 12 \]

### Step 8: Check the solution
Substitute \(x = 12\) back into the original equation:
\[ 0.5(12) + \frac{12}{3} = 0.25(12) + 7 \]
Calculating both sides:
Left side:
\[ 0.5 \times 12 + \frac{12}{3} = 6 + 4 = 10 \]
Right side:
\[ 0.25 \times 12 + 7 = 3 + 7 = 10 \]
Since both sides are equal, our solution is verified.

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{12} \]