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If 10 x+5y=-8 and -7x-10y=-9 are true equations, what would be the value of 3x-5y ? Answer:

If 10x+5y=−8 10 x+5 y=-8 and −7x−10y=−9 -\mathbf{7 x}-\mathbf{1 0 y}=-9 are true equations, what would be the value of 3x−5y \mathbf{3 x}-\mathbf{5 y} ?\newlineAnswer:

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Q. If 10x+5y=−8 10 x+5 y=-8 and −7x−10y=−9 -\mathbf{7 x}-\mathbf{1 0 y}=-9 are true equations, what would be the value of 3x−5y \mathbf{3 x}-\mathbf{5 y} ?\newlineAnswer:
  1. Given Equations: We are given two equations:\newline10x+5y=−810x + 5y = -8 (Equation 11)\newline−7x−10y=−9-7x - 10y = -9 (Equation 22)\newlineWe need to find the value of 3x−5y3x - 5y.\newlineFirst, let's try to eliminate one of the variables by multiplying Equation 22 by 0.50.5 to make the coefficients of yy the same.
  2. Eliminate Variable: Multiply Equation 22 by 0.50.5: \newline0.5×(−7x−10y)=0.5×(−9)0.5 \times (-7x - 10y) = 0.5 \times (-9)\newlineThis gives us:\newline−3.5x−5y=−4.5 (Equation 3)-3.5x - 5y = -4.5 \text{ (Equation 3)}\newlineNow we have the yy coefficients the same in both Equation 11 and Equation 33.
  3. Add Equations: Add Equation 11 and Equation 33 to eliminate yy:(10x+5y)+(−3.5x−5y)=−8+(−4.5)(10x + 5y) + (-3.5x - 5y) = -8 + (-4.5)This simplifies to:10x−3.5x=−8−4.510x - 3.5x = -8 - 4.56.5x=−12.56.5x = -12.5Now we can solve for xx.
  4. Solve for x: Divide both sides by 6.56.5 to solve for xx: \newline6.5x6.5=−12.56.5\frac{6.5x}{6.5} = \frac{-12.5}{6.5}\newlinex=−12.56.5x = \frac{-12.5}{6.5}\newlinex=−1.92307692308 (approximately)x = -1.92307692308 \text{ (approximately)}\newlineNow we have the value of xx.
  5. Solve for y: Substitute the value of xx back into Equation 11 to solve for yy:10(−1.92307692308)+5y=−810(-1.92307692308) + 5y = -8−19.2307692308+5y=−8-19.2307692308 + 5y = -8Add 19.2307692308 to both sides to solve for y:\text{Add } 19.2307692308 \text{ to both sides to solve for } y:5y=−8+19.23076923085y = -8 + 19.23076923085y=11.23076923085y = 11.2307692308Now divide by 5 to get the value of y:\text{Now divide by } 5 \text{ to get the value of } y:y=11.23076923085y = \frac{11.2307692308}{5}y=2.24615384616 (approximately)y = 2.24615384616 \text{ (approximately)}
  6. Find 3x−5y3x - 5y: Now that we have both xx and yy, we can find the value of 3x−5y3x - 5y:3(−1.92307692308)−5(2.24615384616)3(-1.92307692308) - 5(2.24615384616)=−5.76923076924−11.2307692308= -5.76923076924 - 11.2307692308=−17.00000000004 (approximately)= -17.00000000004 \text{ (approximately)}However, we should expect an exact value since we are dealing with linear equations. There might be a rounding error due to the approximation of xx and yy. Let's re-evaluate the calculations without rounding.

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