We need to solve the system of equations:

1. $2x + 5y = 8$
2. $4x + y = 7$

Step 1: Solve one equation for one variable

From the second equation, $4x + y = 7$, solve for $y$: $y = 7 - 4x$

Step 2: Substitute $y = 7 - 4x$ into the first equation

Substitute $y$ into $2x + 5y = 8$: $2x + 5(7 - 4x) = 8$ Simplify: $2x + 35 - 20x = 8$ $-18x + 35 = 8$ $-18x = -27$ $x = \frac{27}{18} = \frac{3}{2}$

Step 3: Solve for $y$

Substitute $x = \frac{3}{2}$ into $y = 7 - 4x$: $y = 7 - 4\left(\frac{3}{2}\right)$ $y = 7 - 6 = 1$

Step 4: Write the solution

The solution is: $(x, y) = \left(\frac{3}{2}, 1\right)$

Step 5: Verify the solution

Substitute $x = \frac{3}{2}$ and $y = 1$ into both equations:

1. $2x + 5y = 8$: $2\left(\frac{3}{2}\right) + 5(1) = 3 + 5 = 8$ True.
2. $4x + y = 7$: $4\left(\frac{3}{2}\right) + 1 = 6 + 1 = 7$ True.

The solution is correct.

Final Answer:

A. There is one solution. The solution of the system is $\left(\frac{3}{2}, 1\right)$.

Follow-Up Questions:

1. How would the solution change if one of the equations was altered?
2. What happens when the system of equations has no solution?
3. Can you express the system in matrix form and solve it using matrix methods?
4. How do you identify whether a system is consistent, inconsistent, or dependent?
5. What are alternative methods for solving a system of linear equations?

Tip: Always verify solutions by substituting back into the original equations.