Find the co-ordinates of the point of intersection of the straight lines
`3x-5y+5=0, 2x+3y-22=0`

Answer

To find the coordinates of the point of intersection of the straight lines given by the equations \(3x - 5y + 5 = 0\) and \(2x + 3y - 22 = 0\), we will follow these steps:

### Step 1: Write down the equations
The equations of the lines are:
1. \(3x - 5y + 5 = 0\) (Equation 1)
2. \(2x + 3y - 22 = 0\) (Equation 2)

### Step 2: Rearrange the equations
We can rearrange both equations to express \(y\) in terms of \(x\).

From Equation 1:
\[ 3x - 5y + 5 = 0 \implies 5y = 3x + 5 \implies y = \frac{3x + 5}{5} \]

From Equation 2:
\[ 2x + 3y - 22 = 0 \implies 3y = 22 - 2x \implies y = \frac{22 - 2x}{3} \]

### Step 3: Set the equations for \(y\) equal to each other
Now we can set the two expressions for \(y\) equal to find \(x\):
\[ \frac{3x + 5}{5} = \frac{22 - 2x}{3} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 3(3x + 5) = 5(22 - 2x) \]
Expanding both sides:
\[ 9x + 15 = 110 - 10x \]

### Step 5: Combine like terms
Now, we will combine like terms:
\[ 9x + 10x = 110 - 15 \implies 19x = 95 \]

### Step 6: Solve for \(x\)
Now we can solve for \(x\):
\[ x = \frac{95}{19} = 5 \]

### Step 7: Substitute \(x\) back to find \(y\)
Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). We'll use Equation 1:
\[ 3(5) - 5y + 5 = 0 \implies 15 - 5y + 5 = 0 \implies 20 - 5y = 0 \]
\[ 5y = 20 \implies y = \frac{20}{5} = 4 \]

### Step 8: Write the coordinates of the intersection point
The coordinates of the point of intersection are:
\[ (x, y) = (5, 4) \]

### Final Answer:
The point of intersection of the lines is \((5, 4)\).

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