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If 2x-5y=7 and -9x-2y=10 are true equations, what would be the value of -7x-7y ? Answer:

If 2x−5y=7 \mathbf{2 x}-\mathbf{5 y}=\mathbf{7} and −9x−2y=10 -\mathbf{9 x}-\mathbf{2 y}=\mathbf{1 0} are true equations, what would be the value of −7x−7y -7 x-7 y ?\newlineAnswer:

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Q. If 2x−5y=7 \mathbf{2 x}-\mathbf{5 y}=\mathbf{7} and −9x−2y=10 -\mathbf{9 x}-\mathbf{2 y}=\mathbf{1 0} are true equations, what would be the value of −7x−7y -7 x-7 y ?\newlineAnswer:
  1. Solve for x: Solve the first equation for x.\newlineWe have 2x−5y=72x - 5y = 7. To solve for x, we add 5y5y to both sides and then divide by 22.\newline2x=5y+72x = 5y + 7\newlinex=5y+72x = \frac{5y + 7}{2}
  2. Substitute xx in second equation: Substitute the expression for xx into the second equation.\newlineWe have −9x−2y=10-9x - 2y = 10. Substitute xx with (5y+7)/2(5y + 7) / 2.\newline−9((5y+7)/2)−2y=10-9((5y + 7) / 2) - 2y = 10
  3. Simplify the equation: Simplify the equation.\newlineDistribute −9-9 to both terms inside the parentheses.\newline−9×5y2−9×72−2y=10-9 \times \frac{5y}{2} - 9 \times \frac{7}{2} - 2y = 10\newline−45y2−632−2y=10-\frac{45y}{2} - \frac{63}{2} - 2y = 10
  4. Combine like terms: Combine like terms.\newlineTo combine the yy terms, we need a common denominator. Multiply −2y-2y by 22\frac{2}{2} to get −4y2-\frac{4y}{2}.\newline−45y2−4y2−632=10-\frac{45y}{2} - \frac{4y}{2} - \frac{63}{2} = 10\newline(−45y−4y)/2−632=10\left(-45y - 4y\right) / 2 - \frac{63}{2} = 10\newline−49y2−632=10-\frac{49y}{2} - \frac{63}{2} = 10
  5. Eliminate fractions: Multiply the entire equation by 22 to eliminate the fractions.\newline2(−49y/2−63/2)=2×102(-49y/2 - 63/2) = 2 \times 10\newline−49y−63=20-49y - 63 = 20
  6. Solve for y: Add 6363 to both sides to solve for y.\newline−49y=20+63-49y = 20 + 63\newline−49y=83-49y = 83
  7. Find yy: Divide both sides by −49-49 to find yy.y=83−49y = \frac{83}{-49}y=−8349y = -\frac{83}{49}
  8. Substitute yy in xx: Substitute the value of yy back into the expression for xx.
    x=5y+72x = \frac{5y + 7}{2}
    x=5(−8349)+72x = \frac{5(-\frac{83}{49}) + 7}{2}
  9. Simplify xx expression: Simplify the expression for xx.
    x=−41549+7)/2x = \frac{-415}{49} + 7) / 2
    x=(−41549+34349)/2x = \left(\frac{-415}{49} + \frac{343}{49}\right) / 2
    x=(−7249)/2x = \left(\frac{-72}{49}\right) / 2
    x=−7298x = \frac{-72}{98}
    x=−3649x = \frac{-36}{49}
  10. Calculate −7x−7y-7x - 7y: Calculate the value of −7x−7y-7x - 7y.\(-7x - 77y = −7-7\left(-\frac{3636}{4949}\right) - 77\left(-\frac{8383}{4949}\right)
  11. Simplify expression: Simplify the expression.\newline−7x−7y=25249−58149-7x - 7y = \frac{252}{49} - \frac{581}{49}\newline−7x−7y=252−58149-7x - 7y = \frac{252 - 581}{49}\newline−7x−7y=−32949-7x - 7y = -\frac{329}{49}

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