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Solve the following equations: a) 3x + 2y = 11 and 2x − y = 1. b) If 5^x = 243, find x.

Mathematics

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2 parts
a

x=137,y=197x = \frac{13}{7}, y = \frac{19}{7}

b

x≈3.41x \approx 3.41

Step-by-step solution

7 steps

Okay, let's tackle these questions.

Part a· 3 steps

Solve the simultaneous equations

3x+2y=11(1)3x + 2y = 11 \quad (1) 2x−y=1(2)2x - y = 1 \quad (2)

  1. Multiply equation (2) by 2 to eliminate yy.

    2(2x−y)=2(1)2(2x - y) = 2(1) 4x−2y=2(3)4x - 2y = 2 \quad (3)

  2. Add equation (1) and equation (3).

    (3x+2y)+(4x−2y)=11+2(3x + 2y) + (4x - 2y) = 11 + 2 7x=137x = 13 x=137x = \frac{13}{7}

  3. Substitute the value of xx into equation (2) to find yy.

    2(137)−y=12\left(\frac{13}{7}\right) - y = 1 267−y=1\frac{26}{7} - y = 1 y=267−1y = \frac{26}{7} - 1 y=267−77y = \frac{26}{7} - \frac{7}{7} y=197y = \frac{19}{7}

    The solution is:

    x=137,y=197\boxed{x = \frac{13}{7}, y = \frac{19}{7}}

Part b

x≈3.41x \approx 3.41

If 5x=2435^x = 243, find xx.

  1. Take the logarithm of both sides of the equation.

    log⁡(5x)=log⁡(243)\log(5^x) = \log(243)

  2. Use the logarithm property log⁡(ab)=blog⁡(a)\log(a^b) = b \log(a) to bring xx down.

    xlog⁡(5)=log⁡(243)x \log(5) = \log(243)

  3. Solve for xx.

    x=log⁡(243)log⁡(5)x = \frac{\log(243)}{\log(5)}

  4. Calculate the value using a calculator and round to two decimal places as per the instructions.

    x≈2.3856060.698970x \approx \frac{2.385606}{0.698970} x≈3.4130x \approx 3.4130 x≈3.41x \approx 3.41

    The value of xx is:

    x≈3.41\boxed{x \approx 3.41}
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