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Solve the simultaneous equations: 2x + y = 7x − y = 1.

Mathematics

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Answer

x=29,y=59x = \frac{2}{9}, y = \frac{5}{9}

Step-by-step solution

5 steps

  1. Separate the given expression into two distinct equations.

    The problem states 2x+y=7x−y=12x + y = 7x - y = 1. This can be broken down into:
    Equation (1): 2x+y=12x + y = 1
    Equation (2): 7x−y=17x - y = 1

  2. Add Equation (1) and Equation (2) to eliminate yy.

    (2x+y)+(7x−y)=1+1(2x + y) + (7x - y) = 1 + 1 2x+7x+y−y=22x + 7x + y - y = 2 9x=29x = 2

  3. Solve for xx.

    x=29x = \frac{2}{9}

  4. Substitute the value of xx into Equation (1) to solve for yy.

    Using Equation (1): 2x+y=12x + y = 1
    Substitute x=29x = \frac{2}{9}:

    2(29)+y=12\left(\frac{2}{9}\right) + y = 1 49+y=1\frac{4}{9} + y = 1

  5. Solve for yy.

    y=1−49y = 1 - \frac{4}{9}

    To subtract, find a common denominator:

    y=99−49y = \frac{9}{9} - \frac{4}{9} y=59y = \frac{5}{9}

    The solution to the simultaneous equations is x=29,y=59x = \frac{2}{9}, y = \frac{5}{9}.

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