b What does the solution for a system of equations look graphically? When are two lines parallel?
A
a Sometimes
B
b Never
Practice makes perfect
a We are asked to consider the given system of equations and explain whether it will always, sometimes, or never have exactly one solution if g≥ h. The solution for a system of linear equations is the point at which the lines intersect. Let's consider our system.
y=gx+3 y=hx+7
We know that unless a system of linear equations is parallel — if they have the same slope but different y-intercepts — the lines will cross and have exactly one solution. This means that for g>h, there will be only one solution. However, if g=h, the lines will be parallel and will not have a solution. The correct answer is sometimes.
b We are asked to consider the system of equations shown below and explain if g≤ h will make the system always, sometimes or never have infinitely many solutions.
y=gx+3 y=hx+7
We know from Part A that the lines will be parallel and have no solution if g=h. If gcoincidental and have the same equation. Since the equations in our system have different y-intercepts, they will never have infinitely many solutions.
