To solve the given systems of equations analytically and graphically, we will follow these steps:

3) y = 2x^2 - 4x + 5 and y = -2x + 9

Analytical Solution:
Step 1: Set the two equations equal to each other:
2x^2 - 4x + 5 = -2x + 9

Step 2: Rearrange the equation to standard form:
2x^2 - 4x + 2x - 4 = 9 - 5
2x^2 - 2x - 4 = 4

Step 3: Simplify the equation:
2x^2 - 2x - 8 = 0

Step 4: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values a = 2, b = -2, and c = -8, we get:
x = (-(-2) ± √((-2)^2 - 4(2)(-8))) / (2(2))
x = (2 ± √(4 + 64)) / 4
x = (2 ± √68) / 4
x = (2 ± 2√17) / 4
x = (1 ± √17) / 2

Step 5: Substitute the values of x back into one of the original equations to find the corresponding y-values. Let's use the equation y = -2x + 9:
For x = (1 + √17) / 2:
y = -2((1 + √17) / 2) + 9
y = -1 - √17 + 9
y = 8 - √17

For x = (1 - √17) / 2:
y = -2((1 - √17) / 2) + 9
y = -1 + √17 + 9
y = 8 + √17

Therefore, the solutions to the system of equations are:
(x, y) = ((1 + √17) / 2, 8 - √17) and ((1 - √17) / 2, 8 + √17)

Graphical Solution:
To graphically solve the system of equations, we will plot the two equations on a coordinate plane and find the points of intersection.

For the first equation, y = 2x^2 - 4x + 5, we can plot points by substituting different x-values and calculating the corresponding y-values.

For the second equation, y = -2x + 9, we can also plot points by substituting different x-values and calculating the corresponding y-values.

By plotting the points and observing the graph, we can find the points of intersection, which represent the solutions to the system of equations.

4) 3x^2 - y^2 = 1 and x^2 - y^2 = -1

Analytical Solution:
Step 1: Set the two equations equal to each other:
3x^2 - y^2 = 1
x^2 - y^2 = -1

Step 2: Subtract the second equation from the first equation to eliminate the y^2 term:
(3x^2 - y^2) - (x^2 - y^2) = 1 - (-1)
2x^2 = 2
x^2 = 1
x = ±1

Step 3: Substitute the values of x back into one of the original equations to find the corresponding y-values. Let's use the equation x^2 - y^2 = -1:
For x = 1:
1 - y^2 = -1
y^2 = 2
y = ±√2

For x = -1:
(-1) - y^2 = -1
y^2 = 0
y = 0

Therefore, the solutions to the system of equations are:
(x, y) = (1, √2), (1, -√2), (-1, 0)

Graphical Solution:
To graphically solve the system of equations, we will plot the two equations on a coordinate plane and find the points of intersection.

For the first equation, 3x^2 - y^2 = 1, we can plot points by substituting different x-values and calculating the corresponding y-values.

For the second equation, x^2 - y^2 = -1, we can also plot points by substituting different x-values and calculating the corresponding y-values.

By plotting the points and observing the graph, we can find the points of intersection, which represent the solutions to the system of equations.

Question

To solve the given systems of equations analytically and graphically, we will follow these steps:

3) y = 2x^2 - 4x + 5 and y = -2x + 9

Analytical Solution:
Step 1: Set the two equations equal to each other:
2x^2 - 4x + 5 = -2x + 9

Step 2: Rearrange the equation to standard form:
2x^2 - 4x + 2x - 4 = 9 - 5
2x^2 - 2x - 4 = 4

Step 3: Simplify the equation:
2x^2 - 2x - 8 = 0

Step 4: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values a = 2, b = -2, and c = -8, we get:
x = (-(-2) ± √((-2)^2 - 4(2)(-8))) / (2(2))
x = (2 ± √(4 + 64)) / 4
x = (2 ± √68) / 4
x = (2 ± 2√17) / 4
x = (1 ± √17) / 2

Step 5: Substitute the values of x back into one of the original equations to find the corresponding y-values. Let's use the equation y = -2x + 9:
For x = (1 + √17) / 2:
y = -2((1 + √17) / 2) + 9
y = -1 - √17 + 9
y = 8 - √17

For x = (1 - √17) / 2:
y = -2((1 - √17) / 2) + 9
y = -1 + √17 + 9
y = 8 + √17

Therefore, the solutions to the system of equations are:
(x, y) = ((1 + √17) / 2, 8 - √17) and ((1 - √17) / 2, 8 + √17)

Graphical Solution:
To graphically solve the system of equations, we will plot the two equations on a coordinate plane and find the points of intersection.

For the first equation, y = 2x^2 - 4x + 5, we can plot points by substituting different x-values and calculating the corresponding y-values.

For the second equation, y = -2x + 9, we can also plot points by substituting different x-values and calculating the corresponding y-values.

By plotting the points and observing the graph, we can find the points of intersection, which represent the solutions to the system of equations.

4) 3x^2 - y^2 = 1 and x^2 - y^2 = -1

Analytical Solution:
Step 1: Set the two equations equal to each other:
3x^2 - y^2 = 1
x^2 - y^2 = -1

Step 2: Subtract the second equation from the first equation to eliminate the y^2 term:
(3x^2 - y^2) - (x^2 - y^2) = 1 - (-1)
2x^2 = 2
x^2 = 1
x = ±1

Step 3: Substitute the values of x back into one of the original equations to find the corresponding y-values. Let's use the equation x^2 - y^2 = -1:
For x = 1:
1 - y^2 = -1
y^2 = 2
y = ±√2

For x = -1:
(-1) - y^2 = -1
y^2 = 0
y = 0

Therefore, the solutions to the system of equations are:
(x, y) = (1, √2), (1, -√2), (-1, 0)

Graphical Solution:
To graphically solve the system of equations, we will plot the two equations on a coordinate plane and find the points of intersection.

For the first equation, 3x^2 - y^2 = 1, we can plot points by substituting different x-values and calculating the corresponding y-values.

For the second equation, x^2 - y^2 = -1, we can also plot points by substituting different x-values and calculating the corresponding y-values.

By plotting the points and observing the graph, we can find the points of intersection, which represent the solutions to the system of equations.

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