We are given the system of equations:

5x - 10y = 9 \tag{1} -3x + 16y = 5 \tag{2}

To determine how many solutions this system has, we can use the method of substitution or elimination.

Let's use elimination to solve this.

Step 1: Multiply the equations to align the coefficients of $x$ or $y$

We want to eliminate one of the variables by making their coefficients equal. Let's eliminate $x$.

• Multiply equation (1) by 3 and equation (2) by 5 so the coefficients of $x$ will become equal (but with opposite signs):

3(5x - 10y) = 3(9) \quad \Rightarrow \quad 15x - 30y = 27 \tag{3} 5(-3x + 16y) = 5(5) \quad \Rightarrow \quad -15x + 80y = 25 \tag{4}

Step 2: Add the two new equations

Now, add equations (3) and (4) together to eliminate $x$:

$(15x - 30y) + (-15x + 80y) = 27 + 25$ $0x + 50y = 52$ $50y = 52$ $y = \frac{52}{50} = \frac{26}{25}$

Step 3: Substitute $y = \frac{26}{25}$ into one of the original equations

Now that we know $y$, we can substitute it back into one of the original equations to solve for $x$. Using equation (1):

$5x - 10\left(\frac{26}{25}\right) = 9$ $5x - \frac{260}{25} = 9$ $5x - 10.4 = 9$ $5x = 9 + 10.4$ $5x = 19.4$ $x = \frac{19.4}{5} = 3.88$

Conclusion:

The system has one solution, which is approximately $x = 3.88$ and $y = \frac{26}{25} \approx 1.04$.

Would you like me to go into more details or clarify anything further?

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Here are 5 related questions you might find interesting:

1. How do you solve a system of equations using substitution?
2. What is the significance of having exactly one solution for a system of linear equations?
3. How do you recognize if a system of equations has no solution or infinitely many solutions?
4. Can you solve a system of equations using matrices?
5. What is the geometric interpretation of the solution to a system of two linear equations?

Tip: If you ever get stuck in the elimination method, try multiplying the equations by factors that simplify the coefficients and ensure they will cancel out when added or subtracted.