The simultaneous equations `2x+3y=5, 4x+5y=10` have

Options

A. no solution
B. only one solution
C. only two solutions
D. several solutions

Answer

To determine the number of solutions for the simultaneous equations \(2x + 3y = 5\) and \(4x + 5y = 10\), we can follow these steps:

### Step 1: Write the equations in standard form
The given equations are already in the standard form:
1. \(2x + 3y - 5 = 0\) (Equation 1)
2. \(4x + 5y - 10 = 0\) (Equation 2)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For Equation 1: \(a_1 = 2\), \(b_1 = 3\), \(c_1 = -5\)
- For Equation 2: \(a_2 = 4\), \(b_2 = 5\), \(c_2 = -10\)

### Step 3: Calculate the ratios
Now, we will calculate the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \):
- \( \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{3}{5} \)
- \( \frac{c_1}{c_2} = \frac{-5}{-10} = \frac{1}{2} \)

### Step 4: Analyze the ratios
Now we compare the ratios:
- \( \frac{a_1}{a_2} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{3}{5} \)
- \( \frac{c_1}{c_2} = \frac{1}{2} \)

### Step 5: Determine the type of solution
According to the conditions:
- If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), there are no solutions.
- If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), there is exactly one solution.
- If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), there are infinitely many solutions.

In this case:
- \( \frac{a_1}{a_2} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{3}{5} \)
- \( \frac{c_1}{c_2} = \frac{1}{2} \)

Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), we conclude that the system of equations has exactly **one solution**.

### Final Answer
The simultaneous equations \(2x + 3y = 5\) and \(4x + 5y = 10\) have **one solution**.

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