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If 7x-8y=-10 and 10x+4y=6 are true equations, what would be the value of 17 x-4y ? Answer:

If 7x−8y=−10 \mathbf{7 x}-\mathbf{8 y}=-\mathbf{1 0} and 10x+4y=6 \mathbf{1 0 x}+\mathbf{4 y}=\mathbf{6} are true equations, what would be the value of 17x−4y 17 x-4 y ?\newlineAnswer:

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Q. If 7x−8y=−10 \mathbf{7 x}-\mathbf{8 y}=-\mathbf{1 0} and 10x+4y=6 \mathbf{1 0 x}+\mathbf{4 y}=\mathbf{6} are true equations, what would be the value of 17x−4y 17 x-4 y ?\newlineAnswer:
  1. Equations Setup: We have two equations:\newline11) 7x−8y=−107x - 8y = -10\newline22) 10x+4y=610x + 4y = 6\newlineWe need to find the value of 17x−4y17x - 4y. To do this, we can use the method of elimination or substitution. Let's use elimination by multiplying the second equation by 22 to make the coefficient of yy in both equations the same.
  2. Multiply Second Equation: Multiply the second equation by 22: \newline2×(10x+4y)=2×62 \times (10x + 4y) = 2 \times 6 \newlineThis gives us: \newline20x+8y=1220x + 8y = 12 \newlineNow we have a new set of equations: \newline11) 7x−8y=−107x - 8y = -10 \newline22) 20x+8y=1220x + 8y = 12
  3. Add Equations: We can now add the two equations together to eliminate yy:(7x−8y)+(20x+8y)=−10+12(7x - 8y) + (20x + 8y) = -10 + 12This simplifies to:7x+20x=27x + 20x = 2
  4. Solve for x: Combine like terms:\newline27x=227x = 2\newlineNow we need to solve for x:\newlinex=227x = \frac{2}{27}
  5. Substitute xx into Equation: Now that we have the value of xx, we can substitute it back into one of the original equations to find the value of yy. Let's use the first equation:\newline7x−8y=−107x - 8y = -10\newlineSubstitute x=227x = \frac{2}{27} into the equation:\newline7×(227)−8y=−107 \times \left(\frac{2}{27}\right) - 8y = -10
  6. Isolate y Term: Simplify the equation:\newline1427−8y=−10\frac{14}{27} - 8y = -10\newlineTo solve for y, we need to isolate the y term. Let's move 1427\frac{14}{27} to the other side of the equation:\newline−8y=−10−1427-8y = -10 - \frac{14}{27}
  7. Divide by −8-8: To combine the terms on the right side, we need a common denominator. The common denominator for 1010 and 1427\frac{14}{27} is 2727:\(-8y = \left(-\frac{270270}{2727}\right) - \left(\frac{1414}{2727}\right)
  8. Find y Value: Combine the terms on the right side:\newline−8y=−28427-8y = -\frac{284}{27}\newlineNow, divide both sides by −8-8 to solve for y:\newliney=−28427−8y = \frac{-\frac{284}{27}}{-8}
  9. Substitute xx and yy: Simplify the division:\newliney=284(27×8)y = \frac{284}{(27 \times 8)}\newliney=284216y = \frac{284}{216}\newliney=7154y = \frac{71}{54}
  10. Simplify Expression: Now we have the values of xx and yy. We can find the value of 17x−4y17x - 4y by substituting these values into the expression:\newline17x−4y=17×(227)−4×(7154)17x - 4y = 17 \times \left(\frac{2}{27}\right) - 4 \times \left(\frac{71}{54}\right)
  11. Simplify Expression: Now we have the values of xx and yy. We can find the value of 17x−4y17x - 4y by substituting these values into the expression:\newline17x−4y=17×(227)−4×(7154)17x - 4y = 17 \times \left(\frac{2}{27}\right) - 4 \times \left(\frac{71}{54}\right)Simplify the expression:\newline17x−4y=(3427)−(4×7154)17x - 4y = \left(\frac{34}{27}\right) - \left(4 \times \frac{71}{54}\right)\newlineTo subtract these fractions, we need a common denominator. The common denominator for 2727 and 5454 is 5454:\newline17x−4y=(34×254)−(4×7154)17x - 4y = \left(\frac{34 \times 2}{54}\right) - \left(4 \times \frac{71}{54}\right)
  12. Simplify Expression: Now we have the values of xx and yy. We can find the value of 17x−4y17x - 4y by substituting these values into the expression:\newline17x−4y=17×(227)−4×(7154)17x - 4y = 17 \times \left(\frac{2}{27}\right) - 4 \times \left(\frac{71}{54}\right)Simplify the expression:\newline17x−4y=(3427)−(4×7154)17x - 4y = \left(\frac{34}{27}\right) - \left(4 \times \frac{71}{54}\right)To subtract these fractions, we need a common denominator. The common denominator for 2727 and 5454 is 5454:\newline17x−4y=(34×254)−(4×7154)17x - 4y = \left(\frac{34 \times 2}{54}\right) - \left(4 \times \frac{71}{54}\right)Combine the terms:\newline17x−4y=(6854)−(28454)17x - 4y = \left(\frac{68}{54}\right) - \left(\frac{284}{54}\right)\newline17x−4y=68−2845417x - 4y = \frac{68 - 284}{54}
  13. Simplify Expression: Now we have the values of xx and yy. We can find the value of 17x−4y17x - 4y by substituting these values into the expression:\newline17x−4y=17×(227)−4×(7154)17x - 4y = 17 \times (\frac{2}{27}) - 4 \times (\frac{71}{54}) Simplify the expression:\newline17x−4y=(3427)−(4×7154)17x - 4y = (\frac{34}{27}) - (4 \times \frac{71}{54})\newlineTo subtract these fractions, we need a common denominator. The common denominator for 2727 and 5454 is 5454:\newline17x−4y=(34×254)−(4×7154)17x - 4y = (\frac{34 \times 2}{54}) - (4 \times \frac{71}{54}) Combine the terms:\newline17x−4y=(6854)−(28454)17x - 4y = (\frac{68}{54}) - (\frac{284}{54})\newlineyy00 Subtract the numerators:\newlineyy11\newlineSimplify the fraction:\newlineyy22

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