Solve the following equation.Check your result in each case
`(2x-3)/5+(x+3)/4=(4x+1)/7`

Answer

To solve the equation \(\frac{2x-3}{5} + \frac{x+3}{4} = \frac{4x+1}{7}\), we will follow these steps:

### Step 1: Find the LCM of the denominators
The denominators are 5, 4, and 7. The least common multiple (LCM) of these numbers is 140.

### Step 2: Rewrite the equation with the LCM
Multiply each term by 140 to eliminate the denominators:
\[ 140 \cdot \left(\frac{2x-3}{5}\right) + 140 \cdot \left(\frac{x+3}{4}\right) = 140 \cdot \left(\frac{4x+1}{7}\right) \]

### Step 3: Simplify each term
Calculating each term:
\[ 28(2x-3) + 35(x+3) = 20(4x+1) \]

### Step 4: Distribute the multiplication
Distributing the terms gives:
\[ 56x - 84 + 35x + 105 = 80x + 20 \]

### Step 5: Combine like terms
Combine the \(x\) terms and the constant terms on the left side:
\[ (56x + 35x) + (-84 + 105) = 80x + 20 \]
This simplifies to:
\[ 91x + 21 = 80x + 20 \]

### Step 6: Move all \(x\) terms to one side
Subtract \(80x\) from both sides:
\[ 91x - 80x + 21 = 20 \]
This simplifies to:
\[ 11x + 21 = 20 \]

### Step 7: Move constant terms to the other side
Subtract 21 from both sides:
\[ 11x = 20 - 21 \]
This simplifies to:
\[ 11x = -1 \]

### Step 8: Solve for \(x\)
Divide both sides by 11:
\[ x = -\frac{1}{11} \]

### Step 9: Check the solution
Substitute \(x = -\frac{1}{11}\) back into the original equation to verify:
\[ \frac{2(-\frac{1}{11})-3}{5} + \frac{-\frac{1}{11}+3}{4} = \frac{4(-\frac{1}{11})+1}{7} \]

Calculating each side:
1. Left Side:
\[ \frac{-\frac{2}{11}-\frac{33}{11}}{5} + \frac{-\frac{1}{11}+\frac{33}{11}}{4} = \frac{-\frac{35}{11}}{5} + \frac{\frac{32}{11}}{4} \]
\[ = -\frac{7}{11} + \frac{8}{11} = \frac{1}{11} \]

2. Right Side:
\[ \frac{-\frac{4}{11}+1}{7} = \frac{-\frac{4}{11}+\frac{11}{11}}{7} = \frac{\frac{7}{11}}{7} = \frac{1}{11} \]

Since both sides are equal, the solution is verified.

### Final Answer
\[ x = -\frac{1}{11} \]