Get the answers you need from a community of experts on IDNLearn.com. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.
Sagot :
To find the solutions for the system of equations:
[tex]\[ \left\{\begin{array}{r} x^2 + y^2 = 4 \\ x - y = 1 \end{array}\right. \][/tex]
we will solve this step-by-step:
1. Start with the second equation:
[tex]\[ x - y = 1 \][/tex]
2. Express \( x \) in terms of \( y \):
[tex]\[ x = y + 1 \][/tex]
3. Substitute \( x \) from the second equation (into the first equation):
[tex]\[ (y + 1)^2 + y^2 = 4 \][/tex]
4. Expand and simplify the equation:
[tex]\[ (y + 1)^2 + y^2 = 4 \\ (y^2 + 2y + 1) + y^2 = 4 \\ 2y^2 + 2y + 1 = 4 \][/tex]
5. Move all terms to one side to set up a quadratic equation:
[tex]\[ 2y^2 + 2y + 1 - 4 = 0 \\ 2y^2 + 2y - 3 = 0 \][/tex]
6. Solve the quadratic equation using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) :
[tex]\[ a = 2, \; b = 2, \; c = -3 \][/tex]
[tex]\[ y = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} \\ y = \frac{-2 \pm \sqrt{4 + 24}}{4} \\ y = \frac{-2 \pm \sqrt{28}}{4} \\ y = \frac{-2 \pm 2\sqrt{7}}{4} \\ y = \frac{-1 \pm \sqrt{7}}{2} \][/tex]
So we have two possible values for \( y \):
[tex]\[ y_1 = \frac{-1 - \sqrt{7}}{2} \quad \text{and} \quad y_2 = \frac{-1 + \sqrt{7}}{2} \][/tex]
7. Substitute these values back into \( x = y + 1 \) to find the corresponding \( x \) values:
For \( y_1 = \frac{-1 - \sqrt{7}}{2} \):
[tex]\[ x_1 = \left( \frac{-1 - \sqrt{7}}{2} \right) + 1 = \frac{-1 - \sqrt{7} + 2}{2} = \frac{1 - \sqrt{7}}{2} \][/tex]
For \( y_2 = \frac{-1 + \sqrt{7}}{2} \):
[tex]\[ x_2 = \left( \frac{-1 + \sqrt{7}}{2} \right) + 1 = \frac{-1 + \sqrt{7} + 2}{2} = \frac{1 + \sqrt{7}}{2} \][/tex]
Thus, we have the solutions:
[tex]\[ \left( \frac{1 - \sqrt{7}}{2}, \frac{-1 - \sqrt{7}}{2} \right) \quad \text{and} \quad \left( \frac{1 + \sqrt{7}}{2}, \frac{-1 + \sqrt{7}}{2} \right) \][/tex]
Which graph represents the solutions:
To visualize the solutions, you would plot the points \(\left( \frac{1 - \sqrt{7}}{2}, \frac{-1 - \sqrt{7}}{2} \right)\) and \(\left( \frac{1 + \sqrt{7}}{2}, \frac{-1 + \sqrt{7}}{2} \right)\) on the coordinate plane along with the circle \(x^2 + y^2 = 4\) and the line \(x - y = 1\).
These points will lie on the intersection of the circle with radius 2 centered at the origin and the line [tex]\(x - y = 1\)[/tex]. The graphical representation will show where the line intersects the circle, indicating the solutions to the system.
[tex]\[ \left\{\begin{array}{r} x^2 + y^2 = 4 \\ x - y = 1 \end{array}\right. \][/tex]
we will solve this step-by-step:
1. Start with the second equation:
[tex]\[ x - y = 1 \][/tex]
2. Express \( x \) in terms of \( y \):
[tex]\[ x = y + 1 \][/tex]
3. Substitute \( x \) from the second equation (into the first equation):
[tex]\[ (y + 1)^2 + y^2 = 4 \][/tex]
4. Expand and simplify the equation:
[tex]\[ (y + 1)^2 + y^2 = 4 \\ (y^2 + 2y + 1) + y^2 = 4 \\ 2y^2 + 2y + 1 = 4 \][/tex]
5. Move all terms to one side to set up a quadratic equation:
[tex]\[ 2y^2 + 2y + 1 - 4 = 0 \\ 2y^2 + 2y - 3 = 0 \][/tex]
6. Solve the quadratic equation using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) :
[tex]\[ a = 2, \; b = 2, \; c = -3 \][/tex]
[tex]\[ y = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} \\ y = \frac{-2 \pm \sqrt{4 + 24}}{4} \\ y = \frac{-2 \pm \sqrt{28}}{4} \\ y = \frac{-2 \pm 2\sqrt{7}}{4} \\ y = \frac{-1 \pm \sqrt{7}}{2} \][/tex]
So we have two possible values for \( y \):
[tex]\[ y_1 = \frac{-1 - \sqrt{7}}{2} \quad \text{and} \quad y_2 = \frac{-1 + \sqrt{7}}{2} \][/tex]
7. Substitute these values back into \( x = y + 1 \) to find the corresponding \( x \) values:
For \( y_1 = \frac{-1 - \sqrt{7}}{2} \):
[tex]\[ x_1 = \left( \frac{-1 - \sqrt{7}}{2} \right) + 1 = \frac{-1 - \sqrt{7} + 2}{2} = \frac{1 - \sqrt{7}}{2} \][/tex]
For \( y_2 = \frac{-1 + \sqrt{7}}{2} \):
[tex]\[ x_2 = \left( \frac{-1 + \sqrt{7}}{2} \right) + 1 = \frac{-1 + \sqrt{7} + 2}{2} = \frac{1 + \sqrt{7}}{2} \][/tex]
Thus, we have the solutions:
[tex]\[ \left( \frac{1 - \sqrt{7}}{2}, \frac{-1 - \sqrt{7}}{2} \right) \quad \text{and} \quad \left( \frac{1 + \sqrt{7}}{2}, \frac{-1 + \sqrt{7}}{2} \right) \][/tex]
Which graph represents the solutions:
To visualize the solutions, you would plot the points \(\left( \frac{1 - \sqrt{7}}{2}, \frac{-1 - \sqrt{7}}{2} \right)\) and \(\left( \frac{1 + \sqrt{7}}{2}, \frac{-1 + \sqrt{7}}{2} \right)\) on the coordinate plane along with the circle \(x^2 + y^2 = 4\) and the line \(x - y = 1\).
These points will lie on the intersection of the circle with radius 2 centered at the origin and the line [tex]\(x - y = 1\)[/tex]. The graphical representation will show where the line intersects the circle, indicating the solutions to the system.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.
