QuestionJuly 14, 2025

55. ) 4x-y+z=-5 2x+2y+3z=10 5x-2y+6z=1

55. ) 4x-y+z=-5 2x+2y+3z=10 5x-2y+6z=1
55.
 ) 4x-y+z=-5 2x+2y+3z=10 5x-2y+6z=1

Solution
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Answer

x = -1, y = 3, z = 2 Explanation 1. Use elimination to simplify equations Multiply the first equation by 2 and subtract from the second equation to eliminate x: 2(4x - y + z) = 2(-5) gives 8x - 2y + 2z = -10. Subtracting from 2x + 2y + 3z = 10 gives -6x + 4y + z = 20. 2. Eliminate x again Multiply the first equation by 5 and subtract from the third equation: 5(4x - y + z) = 5(-5) gives 20x - 5y + 5z = -25. Subtracting from 5x - 2y + 6z = 1 gives -15x + 3y + z = 26. 3. Solve the new system of two equations Solve -6x + 4y + z = 20 and -15x + 3y + z = 26 simultaneously. Subtract the first from the second: (-15x + 3y + z) - (-6x + 4y + z) = 26 - 20 simplifies to -9x - y = 6, or y = -9x - 6. 4. Substitute y into one of the simplified equations Substitute y = -9x - 6 into -6x + 4y + z = 20: -6x + 4(-9x - 6) + z = 20 simplifies to -42x - 24 + z = 20, giving z = 42x + 44. 5. Solve for x Substitute y = -9x - 6 and z = 42x + 44 into any original equation, e.g., 4x - y + z = -5: 4x - (-9x - 6) + (42x + 44) = -5 simplifies to 55x + 50 = -5, solving gives x = -1. 6. Find y and z Substitute x = -1 back into y = -9x - 6 and z = 42x + 44: y = -9(-1) - 6 = 3 and z = 42(-1) + 44 = 2.

Explanation

1. Use elimination to simplify equations<br /> Multiply the first equation by 2 and subtract from the second equation to eliminate $x$: <br />$2(4x - y + z) = 2(-5)$ gives $8x - 2y + 2z = -10$. Subtracting from $2x + 2y + 3z = 10$ gives $-6x + 4y + z = 20$.<br /><br />2. Eliminate $x$ again<br /> Multiply the first equation by 5 and subtract from the third equation:<br />$5(4x - y + z) = 5(-5)$ gives $20x - 5y + 5z = -25$. Subtracting from $5x - 2y + 6z = 1$ gives $-15x + 3y + z = 26$.<br /><br />3. Solve the new system of two equations<br /> Solve $-6x + 4y + z = 20$ and $-15x + 3y + z = 26$ simultaneously. Subtract the first from the second:<br />$(-15x + 3y + z) - (-6x + 4y + z) = 26 - 20$ simplifies to $-9x - y = 6$, or $y = -9x - 6$.<br /><br />4. Substitute $y$ into one of the simplified equations<br /> Substitute $y = -9x - 6$ into $-6x + 4y + z = 20$:<br />$-6x + 4(-9x - 6) + z = 20$ simplifies to $-42x - 24 + z = 20$, giving $z = 42x + 44$.<br /><br />5. Solve for $x$<br /> Substitute $y = -9x - 6$ and $z = 42x + 44$ into any original equation, e.g., $4x - y + z = -5$:<br />$4x - (-9x - 6) + (42x + 44) = -5$ simplifies to $55x + 50 = -5$, solving gives $x = -1$.<br /><br />6. Find $y$ and $z$<br /> Substitute $x = -1$ back into $y = -9x - 6$ and $z = 42x + 44$:<br />$y = -9(-1) - 6 = 3$ and $z = 42(-1) + 44 = 2$.
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