Solve the system of equations. y = x + 21 y = x^2 + x - 15 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Substitute y: Substitute y from the first equation into the second equation. Since y = x + 21, we can replace y in the second equation with x + 21. This gives us x + 21 = x^2 + x - 15.Solve for x: Now, we need to solve for x. To do this, we set the equation to zero by subtracting x + 21 from both sides. This results in 0 = x^2 + x - 15 - (x + 21), which simplifies to 0 = x^2 - 36.Factor the equation: The equation x^2 - 36 = 0 is a difference of squares and can be factored as (x + 6)(x - 6) = 0.Find x values: Setting each factor equal to zero gives us two possible solutions for x: x + 6 = 0 or x - 6 = 0. Solving these, we get x = -6 or x = 6.Find y values: Now we need to find the corresponding y values for each x. When x = -6, substituting into y = x + 21 gives us y = -6 + 21, which simplifies to y = 15.Final solutions: When x = 6, substituting into y = x + 21 gives us y = 6 + 21, which simplifies to y = 27.We now have two solutions for the system of equations: (-6, 15) and (6, 27). These are the coordinates in exact form.

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Solve the system of equations. $\newline$ $y = x + 21$ $\newline$ $y = x^2 + x - 15$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Algebra 2

Solve a nonlinear system of equations

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Q. Solve the system of equations.

\newline

y = x + 21

\newline

y = x^2 + x - 15

\newline

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

\newline

(______,______)

\newline

(______,______)

Substitute $y$ : Substitute $y$ from the first equation into the second equation. Since $y = x + 21$ , we can replace $y$ in the second equation with $x + 21$ . This gives us $x + 21 = x^2 + x - 15$ .
Solve for x: Now, we need to solve for $x$ . To do this, we set the equation to zero by subtracting $x + 21$ from both sides. This results in $0 = x^2 + x - 15 - (x + 21)$ , which simplifies to $0 = x^2 - 36$ .
Factor the equation: The equation $x^2 - 36 = 0$ is a difference of squares and can be factored as $(x + 6)(x - 6) = 0$ .
Find x values: Setting each factor equal to zero gives us two possible solutions for x: $x + 6 = 0$ or $x - 6 = 0$ . Solving these, we get $x = -6$ or $x = 6$ .
Find y values: Now we need to find the corresponding y values for each x. When $x = -6$ , substituting into $y = x + 21$ gives us $y = -6 + 21$ , which simplifies to $y = 15$ .
Final solutions: When $x = 6$ , substituting into $y = x + 21$ gives us $y = 6 + 21$ , which simplifies to $y = 27$ .
Final solutions: When $x = 6$ , substituting into $y = x + 21$ gives us $y = 6 + 21$ , which simplifies to $y = 27$ .We now have two solutions for the system of equations: $(-6, 15)$ and $(6, 27)$ . These are the coordinates in exact form.