Answer :
Sure, let's analyze the given system of equations step-by-step to understand the greatest possible number of solutions.
The system of equations is:
[tex]\[ \begin{cases} x^2 + y^2 = 8 \\ 8x + 2y = 16 \end{cases} \][/tex]
Step 1: Analyze the Equations
1. The first equation [tex]\(x^2 + y^2 = 8\)[/tex] represents a circle centered at the origin [tex]\((0,0)\)[/tex] with a radius of [tex]\(\sqrt{8} = 2\sqrt{2}\)[/tex].
2. The second equation [tex]\(8x + 2y = 16\)[/tex] can be simplified by dividing everything by 2, resulting in [tex]\(4x + y = 8\)[/tex]. This is the equation of a line.
Step 2: Graphical Perspective
1. A circle can generally intersect a line in at most two points. However, this intersection can result in:
- Two distinct points (indicating two solutions).
- One point if the line is tangent to the circle (indicating a single solution).
- No points if the line does not intersect the circle (indicating no solutions).
Step 3: Solve the System Algebraically
To determine how many solutions there are, we need to solve the system of equations.
1. From the equation [tex]\(4x + y = 8\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[ y = 8 - 4x \][/tex]
2. Substitute this expression for [tex]\(y\)[/tex] in the circle's equation [tex]\(x^2 + y^2 = 8\)[/tex]:
[tex]\[ x^2 + (8 - 4x)^2 = 8 \][/tex]
3. Expand and simplify:
[tex]\[ x^2 + (64 - 64x + 16x^2) = 8 \][/tex]
[tex]\[ 17x^2 - 64x + 64 = 8 \][/tex]
[tex]\[ 17x^2 - 64x + 56 = 0 \][/tex]
4. Solve this quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 17\)[/tex], [tex]\(b = -64\)[/tex], and [tex]\(c = 56\)[/tex]:
[tex]\[ x = \frac{64 \pm \sqrt{(-64)^2 - 4 \cdot 17 \cdot 56}}{2 \cdot 17} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{4096 - 3808}}{34} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{288}}{34} \][/tex]
[tex]\[ x = \frac{64 \pm 12\sqrt{2}}{34} \][/tex]
Simplifying further:
[tex]\[ x = \frac{32 \pm 6\sqrt{2}}{17} \][/tex]
5. Substitute these values of [tex]\(x\)[/tex] back into [tex]\(y = 8 - 4x\)[/tex] to find the corresponding [tex]\(y\)[/tex] values.
6. For [tex]\(x = \frac{32 - 6\sqrt{2}}{17}\)[/tex]:
[tex]\[ y = 8 - 4 \left(\frac{32 - 6\sqrt{2}}{17}\right) = 8 - \frac{128 - 24\sqrt{2}}{17} = \frac{136 - (128 - 24\sqrt{2})}{17} = \frac{8 + 24\sqrt{2}}{17} \][/tex]
7. For [tex]\(x = \frac{32 + 6\sqrt{2}}{17}\)[/tex]:
[tex]\[ y = 8 - 4 \left(\frac{32 + 6\sqrt{2}}{17}\right) = 8 - \frac{128 + 24\sqrt{2}}{17} = \frac{136 - (128 + 24\sqrt{2})}{17} = \frac{8 - 24\sqrt{2}}{17} \][/tex]
Thus, the solutions to the system are:
[tex]\[ \left( \frac{32 - 6\sqrt{2}}{17}, \frac{8 + 24\sqrt{2}}{17} \right) \quad \text{and} \quad \left( \frac{32 + 6\sqrt{2}}{17}, \frac{8 - 24\sqrt{2}}{17} \right) \][/tex]
Conclusion:
As the system has two distinct solutions, the greatest possible number of solutions for this system of equations is [tex]\(\boxed{2}\)[/tex].
The system of equations is:
[tex]\[ \begin{cases} x^2 + y^2 = 8 \\ 8x + 2y = 16 \end{cases} \][/tex]
Step 1: Analyze the Equations
1. The first equation [tex]\(x^2 + y^2 = 8\)[/tex] represents a circle centered at the origin [tex]\((0,0)\)[/tex] with a radius of [tex]\(\sqrt{8} = 2\sqrt{2}\)[/tex].
2. The second equation [tex]\(8x + 2y = 16\)[/tex] can be simplified by dividing everything by 2, resulting in [tex]\(4x + y = 8\)[/tex]. This is the equation of a line.
Step 2: Graphical Perspective
1. A circle can generally intersect a line in at most two points. However, this intersection can result in:
- Two distinct points (indicating two solutions).
- One point if the line is tangent to the circle (indicating a single solution).
- No points if the line does not intersect the circle (indicating no solutions).
Step 3: Solve the System Algebraically
To determine how many solutions there are, we need to solve the system of equations.
1. From the equation [tex]\(4x + y = 8\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[ y = 8 - 4x \][/tex]
2. Substitute this expression for [tex]\(y\)[/tex] in the circle's equation [tex]\(x^2 + y^2 = 8\)[/tex]:
[tex]\[ x^2 + (8 - 4x)^2 = 8 \][/tex]
3. Expand and simplify:
[tex]\[ x^2 + (64 - 64x + 16x^2) = 8 \][/tex]
[tex]\[ 17x^2 - 64x + 64 = 8 \][/tex]
[tex]\[ 17x^2 - 64x + 56 = 0 \][/tex]
4. Solve this quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 17\)[/tex], [tex]\(b = -64\)[/tex], and [tex]\(c = 56\)[/tex]:
[tex]\[ x = \frac{64 \pm \sqrt{(-64)^2 - 4 \cdot 17 \cdot 56}}{2 \cdot 17} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{4096 - 3808}}{34} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{288}}{34} \][/tex]
[tex]\[ x = \frac{64 \pm 12\sqrt{2}}{34} \][/tex]
Simplifying further:
[tex]\[ x = \frac{32 \pm 6\sqrt{2}}{17} \][/tex]
5. Substitute these values of [tex]\(x\)[/tex] back into [tex]\(y = 8 - 4x\)[/tex] to find the corresponding [tex]\(y\)[/tex] values.
6. For [tex]\(x = \frac{32 - 6\sqrt{2}}{17}\)[/tex]:
[tex]\[ y = 8 - 4 \left(\frac{32 - 6\sqrt{2}}{17}\right) = 8 - \frac{128 - 24\sqrt{2}}{17} = \frac{136 - (128 - 24\sqrt{2})}{17} = \frac{8 + 24\sqrt{2}}{17} \][/tex]
7. For [tex]\(x = \frac{32 + 6\sqrt{2}}{17}\)[/tex]:
[tex]\[ y = 8 - 4 \left(\frac{32 + 6\sqrt{2}}{17}\right) = 8 - \frac{128 + 24\sqrt{2}}{17} = \frac{136 - (128 + 24\sqrt{2})}{17} = \frac{8 - 24\sqrt{2}}{17} \][/tex]
Thus, the solutions to the system are:
[tex]\[ \left( \frac{32 - 6\sqrt{2}}{17}, \frac{8 + 24\sqrt{2}}{17} \right) \quad \text{and} \quad \left( \frac{32 + 6\sqrt{2}}{17}, \frac{8 - 24\sqrt{2}}{17} \right) \][/tex]
Conclusion:
As the system has two distinct solutions, the greatest possible number of solutions for this system of equations is [tex]\(\boxed{2}\)[/tex].
