Determine whether the points (-3, -5) OR (3,1) are solutions to the system of equations. Analyze each point individually when detailing your answer. In order to receive full points, you’ll need to be detailed and explain with complete sentences your mathematical reasoning. (10 points)

a) Is (-3, -5) a solution to the system?

b) Is (3,1) a solution to the system?

a. Yes, (-3 , -5) is a solution to the system because the graphs of the two equations intersected at it

b. No, (3 , 1) is not a solution to the system because the graphs of the equations did not intersect at it, they intersected at point (1 , 3)

Step-by-step explanation:

The graph has a line which represents an linear equation y = mx + b, where m is the slope of the line and b is the y-intercept (value y at x = 0)

∵ The line intersects the y-axis at point (0 , 1)

∴ b = 1

∵ The line passes through points (0 , 1) and (1 , 3)

∵ m = Δy/Δx

∴ m =

∴ m = 2

The equation of the line is y = 2 x + 1 ⇒ (1)

The graph has a downward parabola which represents a quadratic equation y = ax² + bx + c, where

The quadratic equation is y = -x² + 4 ⇒ (2)

The solutions of the system of equations are the intersection points between the line and the parabola

∵ The points of intersections are (1 , 3) and (-3 , -5)

∴ The solutions of the system of equations are (1 , 3) and (-3 , -5)

Let us verify them algebraically

Equate equations (1) and (2)

∵ 2x + 1 = -x² + 4

- Add x² to both sides

∴ x² + 2x + 1 = 4

- Subtract 4 from both sides

∴ x² + 2x - 3 = 0

- Factorize it into two factors

∴ (x + 3)(x - 1) = 0

- Equate each factor by 0

∵ x + 3 = 0

- subtract 3 from both sides

∴ x = -3

OR

∵ x - 1 = 0

- Add 1 to both sides

∴ x = 1

Substitute the values of x in equation (2) to find the values of y

∵ y = 2(-3) + 1

∴ y = -6 + 1

∴ y = -5

OR

∵ y = 2(1) + 1

∴ y = 2 + 1

∴ y = 3

∴ The solutions are (-3 , -5) and (1 , 3)

Lets answer the questions in a and b

a. Yes, (-3 , -5) is a solution to the system because the graphs of the two equations intersected at it

b. No, (3 , 1) is not a solution to the system because the graphs of the equations did not intersect at it, they intersected at point (1 , 3)

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step-by-step explanation:

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