Solve the system of equations. x^2 + y^2 = 34 x = 4y - 17 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Substitute x: Substitute x from the second equation into the first equation. x = 4y - 17 x^2 + y^2 = 34 (4y - 17)^2 + y^2 = 34Expand and simplify: Expand the squared term and simplify the equation. (4y - 17)^2 + y^2 = 34 16y^2 - 136y + 289 + y^2 = 34 17y^2 - 136y + 289 - 34 = 0 17y^2 - 136y + 255 = 0Factor quadratic equation: Factor the quadratic equation. 17y^2 - 136y + 255 = 0 (17y - 15)(y - 17) = 0Solve for y: Solve for y by setting each factor equal to zero. 17y - 15 = 0 or y - 17 = 0 y = 15/17 or y = 17Substitute y into x: Substitute y back into x = 4y - 17 to find the corresponding x values. For y = 15/17: x = 4(15/17) - 17 x = 60/17 - 17 x = 60/17 - 289/17 x = -229/17Find x values: For y = 17: x = 4(17) - 17 x = 68 - 17 x = 51Write coordinates: Write the coordinates in exact form. First Coordinate: (-229/17, 15/17) Second Coordinate: (51, 17)

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Solve the system of equations. $\newline$ $x^2 + y^2 = 34$ $\newline$ $x = 4y - 17$ $\newline$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Math Problems

Algebra 2

Solve a system of linear and quadratic equations: circles

Full solution

Q. Solve the system of equations.

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x^2 + y^2 = 34

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x = 4y - 17

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

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(______,______)

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(______,______)

Substitute $x$ : Substitute $x$ from the second equation into the first equation. $\newline$ $x = 4y - 17$ $\newline$ $x^2 + y^2 = 34$ $\newline$ $(4y - 17)^2 + y^2 = 34$
Expand and simplify: Expand the squared term and simplify the equation. $\newline$ $(4y - 17)^2 + y^2 = 34$ $\newline$ $16y^2 - 136y + 289 + y^2 = 34$ $\newline$ $17y^2 - 136y + 289 - 34 = 0$ $\newline$ $17y^2 - 136y + 255 = 0$
Factor quadratic equation: Factor the quadratic equation. $\newline$ $17y^2 - 136y + 255 = 0$ $\newline$ $(17y - 15)(y - 17) = 0$
Solve for y: Solve for y by setting each factor equal to zero. $\newline$ $17y - 15 = 0$ or $y - 17 = 0$ $\newline$ $y = \frac{15}{17}$ or $y = 17$
Substitute $y$ into $x$ : Substitute $y$ back into $x = 4y - 17$ to find the corresponding $x$ values. $\newline$ For $y = \frac{15}{17}$ : $\newline$ $x = 4\left(\frac{15}{17}\right) - 17$ $\newline$ $x = \frac{60}{17} - 17$ $\newline$ $x = \frac{60}{17} - \frac{289}{17}$ $\newline$ $x = -\frac{229}{17}$
Find $x$ values: For $y = 17$ : $x = 4(17) - 17$ $x = 68 - 17$ $x = 51$
Write coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(-\frac{229}{17}, \frac{15}{17})$ $\newline$ Second Coordinate: $(51, 17)$