Solve the system of equations y = 4x - 2 and y = x + 3.

Step-by-step solution

1. 1

   Notice both equations are already solved for $y$. That makes substitution trivial: whatever $x$ is, the two right-hand sides must produce the same $y$, so they equal each other. No elimination or matrix work is needed.

2. 2

   Set the right-hand sides equal and collect the $x$ terms.

   $4x-2=x+3\ \Longrightarrow\ 4x-x=3+2\ \Longrightarrow\ 3x=5\ \Longrightarrow\ x=\frac{5}{3}$

3. 3

   Substitute back to get $y$. Using the simpler line $y=x+3$:

   $y=\frac{5}{3}+3=\frac{5}{3}+\frac{9}{3}=\frac{14}{3}$

4. 4

   Confirm with the other equation. The point must satisfy $y=4x-2$ as well:

   $4\left(\frac{5}{3}\right)-2=\frac{20}{3}-\frac{6}{3}=\frac{14}{3}\ \checkmark$

   Both lines give the same $y$, so $\left(\tfrac53,\tfrac{14}3\right)$ is the intersection point.

5. 5

   Interpret the answer graphically. The gradients $4$ and $1$ are different, so the lines are not parallel and there is exactly one intersection — which is why a single point, and not a line or the empty set, is the correct form of the answer. In decimals it sits at about $(1.67,\,4.67)$, which is what a graphing tool would show.