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Solve the following system of linear equations: 2y + 3x = 7 and 4x + 3y = 15.

Mathematics

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Answer

x=−9,y=17x=-9, y=17

Step-by-step solution

6 steps

  1. Rearrange the given equations into the standard form Ax+By=CAx + By = C.

    The given system of equations is:

    2y+3x=7  ⟹  3x+2y=7(1)2y + 3x = 7 \implies 3x + 2y = 7 \quad \text{(1)} 4x+3y=15(2)4x + 3y = 15 \quad \text{(2)}

  2. Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of yy equal.

    Multiply equation (1) by 3:

    3(3x+2y)=3(7)3(3x + 2y) = 3(7) 9x+6y=21(3)9x + 6y = 21 \quad \text{(3)}

    Multiply equation (2) by 2:

    2(4x+3y)=2(15)2(4x + 3y) = 2(15) 8x+6y=30(4)8x + 6y = 30 \quad \text{(4)}

  3. Subtract equation (3) from equation (4) to eliminate yy.

    (8x+6y)−(9x+6y)=30−21(8x + 6y) - (9x + 6y) = 30 - 21 8x−9x+6y−6y=98x - 9x + 6y - 6y = 9 −x=9-x = 9

  4. Solve for xx.

    x=−9x = -9

  5. Substitute the value of xx into equation (1) to solve for yy.

    3(−9)+2y=73(-9) + 2y = 7 −27+2y=7-27 + 2y = 7

  6. Solve for yy.

    2y=7+272y = 7 + 27 2y=342y = 34 y=342y = \frac{34}{2} y=17y = 17

    The solution to the system of equations is:

    x=−9,y=17x=-9, y=17
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