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Solve the system of equations. $\newline$ $y = x^2 + 14x - 35$ $\newline$ $y = 6x + 13$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Solve a nonlinear system of equations

Full solution

Q. Solve the system of equations.

\newline

y = x^2 + 14x - 35

\newline

y = 6x + 13

\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 $x^2 + 14x - 35 = 6x + 13$ .
Subtract and Simplify: Subtract $6x + 13$ from both sides to set the equation to zero. This gives us $x^2 + 14x - 6x - 35 - 13 = 0$ , which simplifies to $x^2 + 8x - 48 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation $x^2 + 8x - 48$ . We are looking for two numbers that multiply to $-48$ and add up to $8$ . These numbers are $12$ and $-4$ . So the factored form is $(x + 12)(x - 4) = 0$ .
Solve for x: Solve for x by setting each factor equal to zero. This gives us $x + 12 = 0$ or $x - 4 = 0$ . Solving these gives us $x = -12$ and $x = 4$ .
Substitute $x = -12$ : Substitute $x = -12$ into one of the original equations to find the corresponding $y$ value. We'll use $y = 6x + 13$ . This gives us $y = 6(-12) + 13$ , which simplifies to $y = -72 + 13$ , resulting in $y = -59$ .
Substitute $x = 4$ : Substitute $x = 4$ into the same equation $y = 6x + 13$ . This gives us $y = 6(4) + 13$ , which simplifies to $y = 24 + 13$ , resulting in $y = 37$ .

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