Find the points of intersection of the line 2x+3y=18 and the cricle `x^(2)+y^(2)=25`.

Answer

To find the points of intersection of the line \(2x + 3y = 18\) and the circle \(x^2 + y^2 = 25\), we will follow these steps:

### Step 1: Express \(y\) in terms of \(x\) from the line equation
Starting with the line equation:
\[ 2x + 3y = 18 \]
We can isolate \(y\):
\[ 3y = 18 - 2x \]
\[ y = \frac{18 - 2x}{3} \]

### Step 2: Substitute \(y\) into the circle equation
Now, substitute the expression for \(y\) into the circle equation:
\[ x^2 + y^2 = 25 \]
Substituting \(y\):
\[ x^2 + \left(\frac{18 - 2x}{3}\right)^2 = 25 \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ x^2 + \frac{(18 - 2x)^2}{9} = 25 \]
Multiply through by 9 to eliminate the fraction:
\[ 9x^2 + (18 - 2x)^2 = 225 \]
Expanding \((18 - 2x)^2\):
\[ 9x^2 + (324 - 72x + 4x^2) = 225 \]
Combine like terms:
\[ 13x^2 - 72x + 324 - 225 = 0 \]
\[ 13x^2 - 72x + 99 = 0 \]

### Step 4: Solve the quadratic equation
Now, we will use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 13\), \(b = -72\), and \(c = 99\):
\[ b^2 - 4ac = (-72)^2 - 4 \cdot 13 \cdot 99 \]
\[ = 5184 - 5148 = 36 \]
Now substituting into the quadratic formula:
\[ x = \frac{72 \pm \sqrt{36}}{2 \cdot 13} \]
\[ = \frac{72 \pm 6}{26} \]
Calculating the two possible values for \(x\):
1. \(x = \frac{78}{26} = 3\)
2. \(x = \frac{66}{26} = \frac{33}{13}\)

### Step 5: Find corresponding \(y\) values
Now, substitute these \(x\) values back into the equation for \(y\):
1. For \(x = 3\):
\[ y = \frac{18 - 2(3)}{3} = \frac{12}{3} = 4 \]
So one point of intersection is \((3, 4)\).

2. For \(x = \frac{33}{13}\):
\[ y = \frac{18 - 2\left(\frac{33}{13}\right)}{3} = \frac{18 - \frac{66}{13}}{3} = \frac{\frac{234 - 66}{13}}{3} = \frac{\frac{168}{13}}{3} = \frac{56}{13} \]
So the second point of intersection is \(\left(\frac{33}{13}, \frac{56}{13}\right)\).

### Final Answer
The points of intersection are:
\[ (3, 4) \quad \text{and} \quad \left(\frac{33}{13}, \frac{56}{13}\right) \]