Answer :
To write an equation of a line in standard form (\(Ax + By = C\)) given a point and the slope, you can follow these steps:
1. **Use the Point-Slope Form**: The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point through which the line passes.
In this case, you have the point \((8, 1)\) and the slope \(3\). Substituting these values into the point-slope form gives us:
\[ y - 1 = 3(x - 8) \]
2. **Expand and Rearrange**: Next, we'll expand the equation and arrange it into the standard form.
\[ y - 1 = 3x - 24 \]
Now, add \(1\) to both sides to isolate y on the left:
\[ y = 3x - 23 \]
3. **Convert to Standard Form**: The standard form of a line is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative.
To convert the equation \(y = 3x - 23\) into standard form, perform these steps:
\[ -3x + y = -23 \]
The standard form typically has the \(x\)-term first. Also, it's conventional for \(A\) to be positive, so if you started with \(-3x + y = -23\), you would multiply through by \(-1\) to achieve that standard:
\[ 3x - y = 23 \]
This is the equation of the line in standard form, where \(A = 3\), \(B = -1\), and \(C = 23\), and all coefficients are integers, as required.
