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Solve the following equations: 1. 2(2a−3) = 5(4+x) 2. 3(5a−3) = 4(2a+9) 3. 5x − (x−10) = 0

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1. 2(2a−3)=5(4+x)2(2a-3) = 5(4+x)

This equation contains two variables, aa and xx. From a single equation, we can express one variable in terms of the other, but we cannot find a unique numerical value for either aa or xx.

  1. Distribute the constants on both sides of the equation.

    2(2a)−2(3)=5(4)+5(x)2(2a) - 2(3) = 5(4) + 5(x) 4a−6=20+5x4a - 6 = 20 + 5x

  2. Isolate aa by adding 6 to both sides.

    4a=20+5x+64a = 20 + 5x + 6 4a=26+5x4a = 26 + 5x

  3. Divide by 4 to express aa in terms of xx.

    a=26+5x4a = \frac{26 + 5x}{4} a=26+5x4\boxed{a = \frac{26 + 5x}{4}}

    (Alternatively, we could express xx in terms of aa: 5x=4a−26  ⟹  x=4a−2655x = 4a - 26 \implies x = \frac{4a - 26}{5})

    2. 3(5a−3)=4(2a+9)3(5a-3) = 4(2a+9)

  4. Distribute the constants on both sides of the equation.

    3(5a)−3(3)=4(2a)+4(9)3(5a) - 3(3) = 4(2a) + 4(9) 15a−9=8a+3615a - 9 = 8a + 36

  5. Subtract $8a$ from both sides of the equation.

    15a−8a−9=3615a - 8a - 9 = 36 7a−9=367a - 9 = 36

  6. Add 9 to both sides of the equation.

    7a=36+97a = 36 + 9 7a=457a = 45

  7. Divide both sides by 7 to solve for aa.

    a=457a = \frac{45}{7} a=457\boxed{a = \frac{45}{7}}

    3. 5x−(x−10)=05x - (x-10) = 0

  8. Distribute the negative sign into the parentheses.

    5x−x+10=05x - x + 10 = 0

  9. Combine like terms.

    4x+10=04x + 10 = 0

  10. Subtract 10 from both sides of the equation.

    4x=−104x = -10

  11. Divide both sides by 4 to solve for xx.

    x=−104x = \frac{-10}{4} x=−52x = -\frac{5}{2} x=−52\boxed{x = -\frac{5}{2}}

    4. 9−(2x−1)=−129 - (2x-1) = -12

  12. Distribute the negative sign into the parentheses.

    9−2x+1=−129 - 2x + 1 = -12

  13. Combine the constant terms on the left side.

    10−2x=−1210 - 2x = -12

  14. Subtract 10 from both sides of the equation.

    −2x=−12−10-2x = -12 - 10 −2x=−22-2x = -22

  15. Divide both sides by -2 to solve for xx.

    x=−22−2x = \frac{-22}{-2} x=11x = 11 x=11\boxed{x = 11}
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