We are given the system of equations:

1. $a = 2b + 3$
2. $a = 4b + 7$

Step 1: Set the equations equal to each other

Since both expressions equal $a$, we can set them equal to each other:

$2b + 3 = 4b + 7$

Step 2: Solve for $b$

First, move all the terms involving $b$ to one side:

$2b - 4b = 7 - 3$

This simplifies to:

$-2b = 4$

Now, solve for $b$:

$b = \frac{4}{-2} = -2$

Step 3: Substitute $b = -2$ into one of the original equations to solve for $a$

Substitute into $a = 2b + 3$:

$a = 2(-2) + 3 = -4 + 3 = -1$

Thus, $b = -2$ and $a = -1$.

Final Answer:

$a = -1, \quad b = -2$

Would you like more details on any of the steps?

Related Questions:

1. How do we solve systems of equations with substitution?
2. Can we use elimination for the same problem?
3. What happens if both equations are identical?
4. How do we verify the solution to a system of equations?
5. How do systems of linear equations differ from nonlinear ones?

Tip:

Always double-check your solution by substituting the values back into both original equations to ensure they satisfy both.