Solving Simultaneous Equations Graphically

When we graph both equations, we see that the lines are parallel (because they have the same slope of 2) and do not intersect. This means there is no point where both equations are satisfied simultaneously. In other words, there is no solution to this system of equations.

When we graph both equations, we see that the lines are parallel (because they have the same slope of -1) and do not intersect. This means there is no point where both equations are satisfied simultaneously. In other words, there is no solution to this system of equations.

When we graph both equations, we see that the lines intersect at one point, meaning there is a unique solution. In this case, the point of intersection is $(1,3)$, so the system has one solution.

When we graph both equations, we see that the lines intersect at one point, meaning there is a unique solution. In this case, the point of intersection is $(4,2)$, so the system has one solution.

If we double the first equation, we get exactly the second equation. When we graph the two equations, the lines overlap perfectly. Every point on the line satisfies both equations. Therefore, the system has infinitely many solutions.

First, let's form the system of equations: $x+y=30$, where $x$ is Emily's age and $y$ is James' age, and $x-y=6$, since Emily is 6 years older than James. Next, solving the system by graphing the equations, we find the point of intersection at $x=18$ and $y=12$. Therefore, Emily is 18 years old, and James is 12 years old.