Solve the system of equations. y = 20x - 34 y = x^2 + 12x - 18 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______)

Set Equations Equal: We have the system of equations: y = 20x - 34 y = x^2 + 12x - 18 To find the intersection points, set the two equations equal to each other. 20x - 34 = x^2 + 12x - 18Rearrange and Identify Quadratic: Rearrange the equation to set it to zero and identify the quadratic equation. 20x - 34 - x^2 - 12x + 18 = 0 -x^2 + 8x - 16 = 0Multiply by -1: Multiply the entire equation by -1 to make the x^2 term positive. x^2 - 8x + 16 = 0Factor Quadratic Equation: Factor the quadratic equation. The factors of 16 that add up to 8 are 4 and 4, so the factored form is: (x - 4)(x - 4) = 0Solve for x: Solve for x. Since both factors are the same, we only have one solution for x: x - 4 = 0 x = 4Find Corresponding y-Value: Find the corresponding y-value by substituting x = 4 into one of the original equations. Using y = 20x - 34: y = 20(4) - 34 y = 80 - 34 y = 46Write Coordinates: Write the coordinates in exact form. Since we only have one solution for x, there is only one intersection point. The coordinates of the intersection point are (4, 46).

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Solve the system of equations. $\newline$ $y = 20x - 34$ $\newline$ $y = x^2 + 12x - 18$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,)

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Math Problems

Algebra 1

Solve a system of linear and quadratic equations

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Q. Solve the system of equations.

\newline

y = 20x - 34

\newline

y = x^2 + 12x - 18

\newline

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

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = 20x - 34$ $\newline$ $y = x^2 + 12x - 18$ $\newline$ To find the intersection points, set the two equations equal to each other. $\newline$ $20x - 34 = x^2 + 12x - 18$
Rearrange and Identify Quadratic: Rearrange the equation to set it to zero and identify the quadratic equation. $\newline$ $20x - 34 - x^2 - 12x + 18 = 0$ $\newline$ $-x^2 + 8x - 16 = 0$
Multiply by $-1$ : Multiply the entire equation by $-1$ to make the $x^2$ term positive. $\newline$ $x^2 - 8x + 16 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ The factors of $16$ that add up to $8$ are $4$ and $4$ , so the factored form is: $\newline$ $(x - 4)(x - 4) = 0$
Solve for x: Solve for x. $\newline$ Since both factors are the same, we only have one solution for $x$ : $\newline$ $x - 4 = 0$ $\newline$ $x = 4$
Find Corresponding y-Value: Find the corresponding y-value by substituting $x = 4$ into one of the original equations. $\newline$ Using $y = 20x - 34$ : $\newline$ $y = 20(4) - 34$ $\newline$ $y = 80 - 34$ $\newline$ $y = 46$
Write Coordinates: Write the coordinates in exact form. $\newline$ Since we only have one solution for $x$ , there is only one intersection point. $\newline$ The coordinates of the intersection point are $(4, 46)$ .