If `10y=7x-4` and `12x+18y=1`. Find the value of `4x+6y` and `8y-x`.

Answer

To solve the given equations \(10y = 7x - 4\) and \(12x + 18y = 1\), and find the values of \(4x + 6y\) and \(8y - x\), we will follow these steps:

### Step 1: Rearranging the first equation
We start with the first equation:
\[ 10y = 7x - 4 \]
To express \(y\) in terms of \(x\), we divide both sides by 10:
\[ y = \frac{7x - 4}{10} \tag{1} \]

### Step 2: Substitute \(y\) into the second equation
Now, we substitute the expression for \(y\) from equation (1) into the second equation \(12x + 18y = 1\):
\[ 12x + 18\left(\frac{7x - 4}{10}\right) = 1 \]

### Step 3: Simplifying the equation
Next, we simplify the equation:
\[ 12x + \frac{18(7x - 4)}{10} = 1 \]
To eliminate the fraction, we can multiply the entire equation by 10:
\[ 10(12x) + 18(7x - 4) = 10 \]
This simplifies to:
\[ 120x + 126x - 72 = 10 \]
Combining like terms gives:
\[ 246x - 72 = 10 \]

### Step 4: Solving for \(x\)
Now, we isolate \(x\):
\[ 246x = 10 + 72 \]
\[ 246x = 82 \]
\[ x = \frac{82}{246} = \frac{1}{3} \tag{2} \]

### Step 5: Finding \(y\)
Now that we have \(x\), we substitute it back into equation (1) to find \(y\):
\[ y = \frac{7\left(\frac{1}{3}\right) - 4}{10} \]
Calculating the numerator:
\[ y = \frac{\frac{7}{3} - 4}{10} = \frac{\frac{7}{3} - \frac{12}{3}}{10} = \frac{-\frac{5}{3}}{10} \]
This simplifies to:
\[ y = -\frac{5}{30} = -\frac{1}{6} \tag{3} \]

### Step 6: Finding \(4x + 6y\)
Now we can find \(4x + 6y\):
\[ 4x + 6y = 4\left(\frac{1}{3}\right) + 6\left(-\frac{1}{6}\right) \]
Calculating each term:
\[ = \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \]

### Step 7: Finding \(8y - x\)
Next, we find \(8y - x\):
\[ 8y - x = 8\left(-\frac{1}{6}\right) - \frac{1}{3} \]
Calculating each term:
\[ = -\frac{8}{6} - \frac{2}{6} = -\frac{10}{6} = -\frac{5}{3} \]

### Final Answers
Thus, the values are:
\[ 4x + 6y = \frac{1}{3} \]
\[ 8y - x = -\frac{5}{3} \]