Solve the system of equations. y = 5x + 34 y = 13x^2 + 31x + 47 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______)

Set Equations Equal: We have the system of equations: y = 5x + 34 y = 13x^2 + 31x + 47 To find the intersection points, we set the two equations equal to each other. 5x + 34 = 13x^2 + 31x + 47Rearrange and Identify Quadratic: Rearrange the equation to set it to zero and identify the quadratic equation. 0 = 13x^2 + 31x + 47 - 5x - 34 0 = 13x^2 + 26x + 13Simplify by Dividing: Notice that all coefficients are divisible by 13, so we can simplify the equation by dividing by 13. 0 = x^2 + 2x + 1 This is a perfect square trinomial.Factor Perfect Square Trinomial: Factor the perfect square trinomial. 0 = (x + 1)(x + 1) 0 = (x + 1)^2Solve for x: Solve for x. Set the factor equal to zero and solve for x. (x + 1) = 0 x = -1Find y-Value: Find the corresponding y-value by substituting x = -1 into either of the original equations. We'll use the first equation for simplicity. y = 5(-1) + 34 y = -5 + 34 y = 29Write Coordinates: Write the coordinates in exact form. Since the quadratic equation had a repeated root, there is only one intersection point. The coordinate is (-1, 29).

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Solve the system of equations. $\newline$ $y = 5x + 34$ $\newline$ $y = 13x^2 + 31x + 47$ $\newline$ 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 = 5x + 34

\newline

y = 13x^2 + 31x + 47

\newline

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

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = 5x + 34$ $\newline$ $y = 13x^2 + 31x + 47$ $\newline$ To find the intersection points, we set the two equations equal to each other. $\newline$ $5x + 34 = 13x^2 + 31x + 47$
Rearrange and Identify Quadratic: Rearrange the equation to set it to zero and identify the quadratic equation. $\newline$ $0 = 13x^2 + 31x + 47 - 5x - 34$ $\newline$ $0 = 13x^2 + 26x + 13$
Simplify by Dividing: Notice that all coefficients are divisible by $13$ , so we can simplify the equation by dividing by $13$ . $0 = x^2 + 2x + 1$ This is a perfect square trinomial.
Factor Perfect Square Trinomial: Factor the perfect square trinomial. $\newline$ $0 = (x + 1)(x + 1)$ $\newline$ $0 = (x + 1)^2$
Solve for x: Solve for x. $\newline$ Set the factor equal to zero and solve for x. $\newline$ $(x + 1) = 0$ $\newline$ $x = -1$
Find y-Value: Find the corresponding y-value by substituting $x = -1$ into either of the original equations. We'll use the first equation for simplicity. $y = 5(-1) + 34$ $y = -5 + 34$ $y = 29$
Write Coordinates: Write the coordinates in exact form. $\newline$ Since the quadratic equation had a repeated root, there is only one intersection point. $\newline$ The coordinate is $(-1, 29)$ .