Final answer:
The system of equations has lines that coincide, have the same slope, and y-intercepts, and therefore have an infinite number of solutions. The statements that the lines are parallel, that the y-intercepts are different, and that there is only one solution are false.
Step-by-step explanation:
When evaluating the system of linear equations 4x + 5y = 15 and 28x + 35y = 105, we need to check the given statements to determine which are true.
- The first statement, or 'a', indicates that the lines coincide. This is true, as the second equation is simply the first equation multiplied by 7, meaning they are the same line.
- Statement 'b', which suggests that the lines are parallel, is false because parallel lines have different y-intercepts, which is not the case here since the lines are identical.
- Statement 'c', stating that the slopes are equal, is true because when reduced to the slope-intercept form both lines will have the same slope.
- Statement 'd' is false, as the y-intercepts are the same due to the fact that the lines coincide.
- Similarly, 'e' is true; they have the same y-intercept.
- Statement 'f' is true because coinciding lines share an infinite number of points.
- The system does not have no solutions (statement 'g'), which is false, because no solutions would be the case for parallel lines, and these are coinciding.
- Lastly, 'h', stating that the system has one solution, is false because there are an infinite number of solutions on coinciding lines.