Final answer:
Any line with the equation y = 4/3x + b, where b is a real number, will be parallel to the line described by the equation 4x - 3y = 1, because parallel lines must have the same slope, which in this case is 4/3.
Step-by-step explanation:
The equation 4x - 3y = 1 represents a straight line on the Cartesian plane. To find an equation of a line parallel to it, we need to understand how the slope works in linear equations. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b is the y-intercept. For two lines to be parallel, they must have the same slope.
To find the slope of the given line, we need to rearrange the equation into slope-intercept form. Subtracting 4x from both sides, we have -3y = -4x + 1. Dividing everything by -3 to solve for y gives us y = 4/3x - 1/3. This means the slope of our line is 4/3.
A parallel line must have the same slope, so its equation will also have the slope of 4/3. However, the y-intercept can be any value. Let's choose a y-intercept of b, thus the equation of a parallel line can be written as y = 4/3x + b, where b can be any real number. For example, if b = 5, the equation of a parallel line would be y = 4/3x + 5.
To summarize, any line with the equation y = 4/3x + b where b is a real number, will be parallel to the line described by the equation 4x - 3y = 1.