What is the point of intersection of the lines 2x + 3y = 5 and 3x - 4y = 10?

Options

A. `(50)/(17),(-5)/(17)`
B. `(-50)/(17),(5)/(17)`
C. `(-10)/(17), (35)/(17)`
D. `(10)/(17),(-35)/(17)`

Answer

To find the point of intersection of the lines given by the equations \(2x + 3y = 5\) and \(3x - 4y = 10\), we can use the substitution or elimination method. Here, we will use the elimination method for clarity.

### Step-by-Step Solution:

**Step 1: Write down the equations.**
We have two equations:
1. \(2x + 3y = 5\) (Equation 1)
2. \(3x - 4y = 10\) (Equation 2)

**Step 2: Make the coefficients of \(x\) or \(y\) the same.**
To eliminate \(x\), we can multiply Equation 1 by 3 and Equation 2 by 2 to make the coefficients of \(x\) the same.

- Multiply Equation 1 by 3:
\[ 3(2x + 3y) = 3(5) \implies 6x + 9y = 15 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 2:
\[ 2(3x - 4y) = 2(10) \implies 6x - 8y = 20 \quad \text{(Equation 4)} \]

**Step 3: Subtract the two new equations.**
Now, we will subtract Equation 4 from Equation 3 to eliminate \(x\):
\[ (6x + 9y) - (6x - 8y) = 15 - 20 \]
This simplifies to:
\[ 9y + 8y = -5 \implies 17y = -5 \]

**Step 4: Solve for \(y\).**
Now, divide both sides by 17:
\[ y = \frac{-5}{17} \]

**Step 5: Substitute \(y\) back into one of the original equations to find \(x\).**
We can substitute \(y\) into Equation 1:
\[ 2x + 3\left(\frac{-5}{17}\right) = 5 \]
This simplifies to:
\[ 2x - \frac{15}{17} = 5 \]
Now, convert 5 into a fraction with a denominator of 17:
\[ 2x - \frac{15}{17} = \frac{85}{17} \]
Now, add \(\frac{15}{17}\) to both sides:
\[ 2x = \frac{85}{17} + \frac{15}{17} = \frac{100}{17} \]
Now, divide both sides by 2:
\[ x = \frac{100}{34} = \frac{50}{17} \]

**Step 6: Write the point of intersection.**
The point of intersection of the two lines is:
\[ \left(\frac{50}{17}, \frac{-5}{17}\right) \]

### Final Answer:
The point of intersection is \(\left(\frac{50}{17}, \frac{-5}{17}\right)\).