We are given the following system of simultaneous equations:

1. $a = a$
2. $a + e = 2a + f$
3. $-4a + 4 + 2f = a + e - 3$
4. $-4f - 1 = e + 3$

Let’s go step-by-step to solve this system:

Step 1: Analyze the first equation

The first equation is trivial: $a = a$ This equation doesn't provide new information, so we can ignore it.

Step 2: Simplify the second equation

The second equation is: $a + e = 2a + f$ Rearranging it to isolate $e$ and $f$ in terms of $a$: $e = a + f$ Now we have an expression for $e$ in terms of $a$ and $f$.

Step 3: Simplify the third equation

The third equation is: $-4a + 4 + 2f = a + e - 3$ Substitute $e = a + f$ from step 2 into this equation: $-4a + 4 + 2f = a + (a + f) - 3$ Simplify the right-hand side: $-4a + 4 + 2f = 2a + f - 3$ Now, combine like terms and isolate $f$: $-4a + 2f + 4 = 2a + f - 3$ Move all terms involving $a$ and $f$ to one side: $-4a - 2a + 2f - f = -3 - 4$ Simplify: $-6a + f = -7$ Thus, we have the equation: $f = 6a - 7$

Step 4: Simplify the fourth equation

The fourth equation is: $-4f - 1 = e + 3$ Substitute $e = a + f$ from step 2 into this equation: $-4f - 1 = (a + f) + 3$ Simplify: $-4f - 1 = a + f + 3$ Rearrange to isolate $f$: $-4f - f = a + 3 + 1$ Simplify: $-5f = a + 4$ Thus: $f = -\frac{a + 4}{5}$

Step 5: Solve for $a$

Now we have two expressions for $f$:

1. $f = 6a - 7$
2. $f = -\frac{a + 4}{5}$

Set these two expressions equal to each other: $6a - 7 = -\frac{a + 4}{5}$ Multiply through by 5 to eliminate the fraction: $5(6a - 7) = -(a + 4)$ Simplify: $30a - 35 = -a - 4$ Combine like terms: $30a + a = 35 - 4$ Simplify: $31a = 31$ Solve for $a$: $a = 1$

Step 6: Find $f$ and $e$

Now that we know $a = 1$, substitute this value into the expressions for $f$ and $e$:

From $f = 6a - 7$: $f = 6(1) - 7 = 6 - 7 = -1$

From $e = a + f$: $e = 1 + (-1) = 0$

Final Solution:

The solution to the system of equations is:

• $a = 1$
• $f = -1$
• $e = 0$

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Would you like further details on any step, or do you have any other questions?

Here are some related questions you could explore:

1. What happens if we have a system with more variables? How would we handle it?
2. How do we check if our solution is correct by substituting back into the original equations?
3. Can we solve a system of equations with more than one solution? When does this happen?
4. What are the methods to solve systems of equations, such as substitution or elimination?
5. How can matrices be used to solve systems of linear equations?

Tip: Always check your solutions by substituting them back into the original equations! This helps ensure that no mistakes were made during the process.