Let the two numbers be $x$ and $y$, where $x$ is the greater number and $y$ is the lesser number. According to the problem:

1. $x = y + 5$ (the greater number is 5 more than the lesser)
2. $3x + 2y = 30$ (three times the greater plus twice the lesser is 30)

Step 1: Substitute $x = y + 5$ into the second equation

$3(y + 5) + 2y = 30$

Step 2: Simplify the equation

$3y + 15 + 2y = 30$ $5y + 15 = 30$

Step 3: Solve for $y$

$5y = 30 - 15$ $5y = 15$ $y = 3$

Step 4: Find $x$

Now, substitute $y = 3$ into $x = y + 5$:

$x = 3 + 5 = 8$

Conclusion:

The two numbers are $x = 8$ and $y = 3$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we solve systems of equations using substitution?
2. What are some alternative methods to solve this problem?
3. Can systems of equations always be solved by substitution?
4. How do we check if our solutions to a system of equations are correct?
5. How do inequalities differ from systems of equations?

Tip: Always check your solution by substituting the values back into the original equations.