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Solve the system of equations. $\newline$ $y = x^2 + 5x + 38$ $\newline$ $y = 19x + 5$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Solve a system of linear and quadratic equations

Full solution

Q. Solve the system of equations.

\newline

y = x^2 + 5x + 38

\newline

y = 19x + 5

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 + 5x + 38$ $\newline$ $y = 19x + 5$ $\newline$ To find the intersection points, set the two equations equal to each other. $\newline$ $x^2 + 5x + 38 = 19x + 5$
Rearrange and Simplify: Rearrange the equation to bring all terms to one side and set it equal to zero. $\newline$ $x^2 + 5x + 38 - 19x - 5 = 0$ $\newline$ $x^2 - 14x + 33 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ In the quadratic equation $ax^2 + bx + c$ , we look for two numbers that multiply to $c$ ( $33$ ) and add up to $b$ ( $-14$ ). $\newline$ The factors of $33$ that add up to $-14$ are $-11$ and $-3$ . $\newline$ $x^2 - 14x + 33 = (x - 11)(x - 3)$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $(x - 11) = 0$ or $(x - 3) = 0$ $\newline$ $x = 11$ or $x = 3$
Find y-Values: Find the corresponding y-values for each x-value by substituting back into either of the original equations. We'll use $y = 19x + 5$ . $\newline$ For $x = 11$ : $\newline$ $y = 19(11) + 5$ $\newline$ $y = 209 + 5$ $\newline$ $y = 214$
Write Coordinates: For $x = 3$ : $\newline$ $y = 19(3) + 5$ $\newline$ $y = 57 + 5$ $\newline$ $y = 62$
Write Coordinates: For $x = 3$ : $\newline$ $y = 19(3) + 5$ $\newline$ $y = 57 + 5$ $\newline$ $y = 62$ Write the coordinates in exact form. $\newline$ First Coordinate: $(11, 214)$ $\newline$ Second Coordinate: $(3, 62)$

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