Questions: Find all solutions of the system of equations algebraically. Write your solutions as coordinate points. y=-3 x^2+26 x-46 x+y=8

Solution Steps

To solve the system of equations algebraically, we can use substitution. First, express \( y \) from the second equation in terms of \( x \). Then, substitute this expression into the first equation to form a quadratic equation in terms of \( x \). Solve the quadratic equation to find the values of \( x \). Finally, substitute these \( x \) values back into the expression for \( y \) to find the corresponding \( y \) values. The solutions will be the coordinate points \((x, y)\).

We start with the system of equations: \[ \begin{align_} y &= -3x^2 + 26x - 46 \quad (1) \\ x + y &= 8 \quad (2) \end{align_} \]

From equation (2), we can express \( y \) in terms of \( x \): \[ y = 8 - x \] Substituting this into equation (1): \[ 8 - x = -3x^2 + 26x - 46 \] Rearranging gives us: \[ 3x^2 - 26x + x + 46 + 8 = 0 \implies 3x^2 - 25x + 54 = 0 \]

Solving the quadratic equation \( 3x^2 - 25x + 54 = 0 \) yields the solutions: \[ x = 3 \quad \text{and} \quad x = 6 \]

Substituting \( x = 3 \) into \( y = 8 - x \): \[ y = 8 - 3 = 5 \] Substituting \( x = 6 \): \[ y = 8 - 6 = 2 \]

Final Answer