Question

I don’t really understand this type of stuff so please help.

Answer 1

For 1 and 2, you plot both lines, and wherever they intersect is the solution to the system. Given the equation of a line, I think the easiest way to plot it is to find two points on the line, then draw a line through them. For example, if
`y=5x-1`, then when
`x=0`, you get
`y=-1`; when
`x=1`, you get
`y=4`. So plot the points (0, -1) and (1, 4), then strike a line through.

1. Notice that dividing both sides of
`2y=10x-2` by 2 returns
`y=5x-1`, same as the first equation. So the system of equations reduces to one equation, which can have an infinite number of solutions. (This is because for any choice of
`x` or
`y`, you can always find a corresponding value for the other variable.)

2. See attached image.
`3x-y=2` is given by the purple line.

For 3-6, you have several options. The two simplest methods of solving them are by substitution or elimination.

3. Like with (1), notice that dividing both sides of the first equation by 2 gives
`x+3y=9`, so there will be an infinite number of solutions.

4. (by substitution) Since
`y=-7x+3`, we can replace
`y` in the second equation:

`-7x+3+7x=10\implies3=10`

but this is false, so there are no solutions to this system.

5. (by substitution) Since
`x=2y+2`, in the first equation we have

`-5(2y+2)+3y=-10y-10+3y=-7y-10=11\implies-7y=21\implies y=-3`

Then back in the second equation we find

`x=2(-3)+2=-6+2=-4`

So (-4, -3) is the only solution here.

6. (by substitution) Notice that the left hand sides of both equations are the same, so we end up with 7 = 12, but this is false, so no solution exists.