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Solve the system of equations: 1) 4x + 3y = 14 2) 2x − 2y = 2

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To solve the system of equations:

  1. 4x+3y=144x + 3y = 14
  2. 2x−2y=22x - 2y = 2

We will use the elimination method.

  1. Multiply the second equation by 2 to make the coefficients of xx match.

    2×(2x−2y)=2×22 \times (2x - 2y) = 2 \times 2 4x−4y=4(Equation 3)4x - 4y = 4 \quad \text{(Equation 3)}

  2. Subtract Equation 3 from Equation 1.

    (4x+3y)−(4x−4y)=14−4(4x + 3y) - (4x - 4y) = 14 - 4 4x+3y−4x+4y=104x + 3y - 4x + 4y = 10 7y=107y = 10

  3. Solve for yy.

    y=107y = \frac{10}{7}

  4. Substitute the value of yy into Equation 2 to solve for xx.

    2x−2(107)=22x - 2\left(\frac{10}{7}\right) = 2 2x−207=22x - \frac{20}{7} = 2

  5. Add 207\frac{20}{7} to both sides of the equation.

    2x=2+2072x = 2 + \frac{20}{7}

    To add the numbers, find a common denominator for 2, which is 147\frac{14}{7}.

    2x=147+2072x = \frac{14}{7} + \frac{20}{7} 2x=14+2072x = \frac{14 + 20}{7} 2x=3472x = \frac{34}{7}

  6. Divide by 2 to solve for xx.

    x=347÷2x = \frac{34}{7} \div 2 x=347×12x = \frac{34}{7} \times \frac{1}{2} x=3414x = \frac{34}{14}

    Simplify the fraction by dividing the numerator and denominator by 2.

    x=177x = \frac{17}{7}

    The solution to the system of equations is:

    x=177x = \boxed{\frac{17}{7}} y=107y = \boxed{\frac{10}{7}}
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