Question

3x - 4y = 6
6x - 8y = 10
The system shown has how many solutions.

Answer 1

Find the slope and y-intercept of each line by rearranging into slope-intercept form \(y = mx+b\). If the lines have equal slope, they are either parallel or coincident. Parallel lines will have a different y-intercept value, and will never intersect, which means no solutions. Coincident means they are the same line, so there are infinitely many solutions. Otherwise, the lines will intersect once, giving 1 solution.

Answer 2

If the equations are in standard form, as in this problem, you can get the slope by dividing the coefficient of \(x\) by the coefficient of \(y\). The number on the right hand side will serve for checking if the intercepts are equal, assuming the slopes turn out to be equal.

\[x-2y = 4\]\[8x-16y = 1\]
Slope of first line is \(1/(-2) = -(1/2)\)
Slope of second line is \(8/(-16) = -(1/2)\)
Slopes are identical, but the values on the right hand side are not, so the lines are parallel and there are no solutions, as seen in this graph: