QuestionJuly 15, 2025

Solve the equation. (5)/(y+2)+(3)/(y-4)=(8)/(y+3) Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is square (Simplify your answer.) B. There is no solution.

Solve the equation. (5)/(y+2)+(3)/(y-4)=(8)/(y+3) Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is square (Simplify your answer.) B. There is no solution.
Solve the equation. (5)/(y+2)+(3)/(y-4)=(8)/(y+3) Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is square (Simplify your answer.) B. There is no solution.

Solution
4.4(180 votes)

Answer

B. There is no solution. Explanation 1. Find a common denominator The common denominator for the fractions is (y+2)(y-4)(y+3). 2. Multiply through by the common denominator Multiply each term by (y+2)(y-4)(y+3) to eliminate the fractions: 5(y-4)(y+3) + 3(y+2)(y+3) = 8(y+2)(y-4). 3. Expand and simplify Expand each term: 5(y^2 - y - 12) + 3(y^2 + 5y + 6) = 8(y^2 - 2y - 8). 4. Combine like terms Simplify the equation: 5y^2 - 5y - 60 + 3y^2 + 15y + 18 = 8y^2 - 16y - 64. 5. Rearrange and solve the quadratic equation Combine and rearrange terms: 8y^2 + 10y - 42 = 8y^2 - 16y - 64. Subtract 8y^2 from both sides: 26y = -22. Solve for y: y = \frac{-22}{26} = \frac{-11}{13}. 6. Check for extraneous solutions Verify if y = \frac{-11}{13} causes any denominator to be zero. It does not.

Explanation

1. Find a common denominator<br /> The common denominator for the fractions is $(y+2)(y-4)(y+3)$.<br /><br />2. Multiply through by the common denominator<br /> Multiply each term by $(y+2)(y-4)(y+3)$ to eliminate the fractions:<br />$5(y-4)(y+3) + 3(y+2)(y+3) = 8(y+2)(y-4)$.<br /><br />3. Expand and simplify<br /> Expand each term:<br />$5(y^2 - y - 12) + 3(y^2 + 5y + 6) = 8(y^2 - 2y - 8)$.<br /><br />4. Combine like terms<br /> Simplify the equation:<br />$5y^2 - 5y - 60 + 3y^2 + 15y + 18 = 8y^2 - 16y - 64$.<br /><br />5. Rearrange and solve the quadratic equation<br /> Combine and rearrange terms:<br />$8y^2 + 10y - 42 = 8y^2 - 16y - 64$.<br /> Subtract $8y^2$ from both sides:<br />$26y = -22$.<br /> Solve for $y$:<br />$y = \frac{-22}{26} = \frac{-11}{13}$.<br /><br />6. Check for extraneous solutions<br /> Verify if $y = \frac{-11}{13}$ causes any denominator to be zero. It does not.
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