Solving Simultaneous Equations Graphically

Both lines have the same gradient of 2, but different y-intercepts (0 and 2). Equal gradients with different intercepts means the lines are parallel, so they never meet.

You can see it algebraically too: setting $2x=2x+2$ gives $0=2$, which is impossible.

Because no point lies on both lines, these simultaneous equations have no solution.

Both lines have the same gradient of $-1$, but different y-intercepts (3 and 5). Equal gradients with different intercepts means the lines are parallel, so they never cross.

Algebraically, setting $-x+3=-x+5$ gives $3=5$, which is impossible.

So no point satisfies both equations: these simultaneous equations have no solution.

The gradients are different ($-1$ and $1$), so the lines are not parallel and must cross exactly once.

Setting the right-hand sides equal: $-x+4=x+2$, so $2=2x$ and $x=1$. Substituting back, $y=1+2=3$.

The lines meet at the single point $(1,3)$, so there is one solution. The pair $x=1$, $y=3$ counts as one solution, not two.

Rearrange each equation into $y=mx+c$ form first: $x-y=2$ becomes $y=x-2$ (gradient 1), and $x+y=6$ becomes $y=-x+6$ (gradient $-1$).

The gradients are different, so the lines cross exactly once. Adding the two original equations eliminates $y$: $2x=8$, so $x=4$, and then $y=2$.

The graphs meet at the single point $(4,2)$, so there is one solution.

Divide every term of the second equation by 2: $2x+2y=2$ becomes $x+y=1$, which is exactly the first equation.

Both equations describe the same line, $y=-x+1$, so the graphs lie on top of each other rather than crossing at a single point.

Every point on that line satisfies both equations, $(0,1)$, $(1,0)$, $(2,-1)$ and so on, so there are infinitely many solutions.

Let x be Emily's age and y be James' age. The two facts give $x+y=30$ and $x-y=6$ (Emily is the older one, so it is her age minus his that equals 6).

On a graph these two lines cross at $(18,12)$. Adding the equations gives the same answer algebraically: $2x=36$, so $x=18$, and substituting into $x+y=30$ gives $y=12$.

Check both facts: $18+12=30$ and $18-12=6$. So Emily is 18 and James is 12.

Note that the 6-year gap is shared between them (each is 3 years either side of the halfway age of 15), so you cannot just add all 6 years to one of them.