Answer :

To solve the system of equations:

[tex]\[ \begin{array}{c} \sqrt{x^2 + y^2} = 25 \\ y - 2x = 5 \end{array} \][/tex]

we will follow these steps:

1. Simplify the First Equation:

The first equation can be simplified by squaring both sides:

[tex]\[ \sqrt{x^2 + y^2} = 25 \implies x^2 + y^2 = 625 \][/tex]

2. Express [tex]\( y \)[/tex] in Terms of [tex]\( x \)[/tex]:

From the second equation, we can express [tex]\( y \)[/tex] as:

[tex]\[ y - 2x = 5 \implies y = 2x + 5 \][/tex]

3. Substitute [tex]\( y \)[/tex] into the First Equation:

Substitute [tex]\( y = 2x + 5 \)[/tex] into [tex]\( x^2 + y^2 = 625 \)[/tex]:

[tex]\[ x^2 + (2x + 5)^2 = 625 \][/tex]

4. Expand and Simplify the Equation:

Expand [tex]\( (2x + 5)^2 \)[/tex]:

[tex]\[ x^2 + (2x + 5)^2 = x^2 + (4x^2 + 20x + 25) = 625 \implies x^2 + 4x^2 + 20x + 25 = 625 \][/tex]

Combine like terms:

[tex]\[ 5x^2 + 20x + 25 = 625 \][/tex]

Subtract 625 from both sides:

[tex]\[ 5x^2 + 20x + 25 - 625 = 0 \implies 5x^2 + 20x - 600 = 0 \][/tex]

Simplify by dividing the entire equation by 5:

[tex]\[ x^2 + 4x - 120 = 0 \][/tex]

5. Solve the Quadratic Equation:

Solve [tex]\( x^2 + 4x - 120 = 0 \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:

Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -120 \)[/tex]:

[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-120)}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 + 480}}{2} = \frac{-4 \pm \sqrt{496}}{2} = \frac{-4 \pm 4\sqrt{31}}{2} = -2 \pm 2\sqrt{31} \][/tex]

So, [tex]\( x = -2 + 2\sqrt{31} \)[/tex] or [tex]\( x = -2 - 2\sqrt{31} \)[/tex].

6. Find the Corresponding [tex]\( y \)[/tex] Values:

Using [tex]\( y = 2x + 5 \)[/tex]:

- For [tex]\( x = -2 + 2\sqrt{31} \)[/tex]:

[tex]\[ y = 2(-2 + 2\sqrt{31}) + 5 = -4 + 4\sqrt{31} + 5 = 1 + 4\sqrt{31} \][/tex]

- For [tex]\( x = -2 - 2\sqrt{31} \)[/tex]:

[tex]\[ y = 2(-2 - 2\sqrt{31}) + 5 = -4 - 4\sqrt{31} + 5 = 1 - 4\sqrt{31} \][/tex]

Therefore, the solutions to the system of equations are:

[tex]\[ \left( -2 + 2\sqrt{31}, 1 + 4\sqrt{31} \right) \quad \text{and} \quad \left( -2 - 2\sqrt{31}, 1 - 4\sqrt{31} \right) \][/tex]