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Solve the system of equations x + 2y = 10 and 3x − y = 5, then calculate 2x - 2y.

Mathematics

ScanSolve AI Answer

Answer

−107-\frac{10}{7}

-1.43

Step-by-step solution

5 steps

Your method for solving the system of equations and calculating the expression is correct. However, rounding the intermediate values for xx and yy to two decimal places introduces a slight error in the final answer. It's generally best to keep values as fractions until the final step to maintain accuracy.

Let's re-evaluate using exact fractions:

Given the system of equations:

x+2y=10(1)x + 2y = 10 \quad (1) 3x−y=5(2)3x - y = 5 \quad (2)
  1. Solve for xx in terms of yy from equation (1).

    x=10−2yx = 10 - 2y
  2. Substitute this expression for xx into equation (2).

    3(10−2y)−y=53(10 - 2y) - y = 5 30−6y−y=530 - 6y - y = 5 30−7y=530 - 7y = 5
  3. Solve for yy.

    30−5=7y30 - 5 = 7y 25=7y25 = 7y y=257y = \frac{25}{7}
  4. Substitute the exact value of yy back into the expression for xx.

    x=10−2(257)x = 10 - 2\left(\frac{25}{7}\right) x=10−507x = 10 - \frac{50}{7}

    To combine, find a common denominator:

    x=707−507x = \frac{70}{7} - \frac{50}{7} x=207x = \frac{20}{7}
  5. Calculate the value of 2x−2y2x - 2y using the exact values of xx and yy.

    2x−2y=2(207)−2(257)2x - 2y = 2\left(\frac{20}{7}\right) - 2\left(\frac{25}{7}\right) 2x−2y=407−5072x - 2y = \frac{40}{7} - \frac{50}{7} 2x−2y=40−5072x - 2y = \frac{40 - 50}{7} 2x−2y=−1072x - 2y = \frac{-10}{7}

    As a decimal, −107≈−1.42857...\frac{-10}{7} \approx -1.42857...
    Rounding to two decimal places, the value is −1.43-1.43.

    Your final answer of −1.42-1.42 is very close, but due to rounding yy to 3.57 and xx to 2.86 in intermediate steps, there's a slight difference.

    The more accurate value is −107\boxed{-\frac{10}{7}} or approximately −1.43\boxed{-1.43}.

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