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Solve the system of equations. $\newline$ $y = x^2 + 23x + 21$ $\newline$ $y = 40x - 9$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Solve a nonlinear system of equations

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Q. Solve the system of equations.

\newline

y = x^2 + 23x + 21

\newline

y = 40x - 9

\newline

Write the coordinates in exact form. Simplify all fractions and radicals.

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: Set the two equations equal to each other since they both equal $y$ . This gives us $x^2 + 23x + 21 = 40x - 9$ .
Rearrange and Solve: Rearrange the equation to set it to zero and solve for $x$ . This means subtracting $40x$ and adding $9$ to both sides, resulting in $x^2 + 23x - 40x + 21 + 9 = 0$ , which simplifies to $x^2 - 17x + 30 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation. We are looking for two numbers that multiply to $30$ and add up to $-17$ . These numbers are $-2$ and $-15$ , so the factored form is $(x - 2)(x - 15) = 0$ .
Solve for x: Solve for x by setting each factor equal to zero. This gives us $x - 2 = 0$ or $x - 15 = 0$ , which means $x = 2$ or $x = 15$ .
Substitute $x$ Values: Substitute $x = 2$ into one of the original equations to find the corresponding $y$ value. Using $y = 40x - 9$ , we get $y = 40(2) - 9$ , which simplifies to $y = 80 - 9$ and then $y = 71$ .
Write Coordinate Points: Substitute $x = 15$ into the same equation to find the corresponding $y$ value. Using $y = 40x - 9$ , we get $y = 40(15) - 9$ , which simplifies to $y = 600 - 9$ and then $y = 591$ .
Write Coordinate Points: Substitute $x = 15$ into the same equation to find the corresponding y value. Using $y = 40x - 9$ , we get $y = 40(15) - 9$ , which simplifies to $y = 600 - 9$ and then $y = 591$ .Write the solution as coordinate points. The coordinate points are $(2, 71)$ and $(15, 591)$ .

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