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Which expression is equivalent to 60(1/4x−2)+20(3/4y−4) (1 point) Responses 15x+15y−6415x+15y−6415 x plus 15 y minus 64 30xy−20030xy−20030 x y minus 200 15x−15y−20015x−15y−20015 x minus 15 y minus 200 15x+15y−200

After simplifying, which expressions are equivalent? Check the three that apply.(3 points) Responses −0.4b+(6b−9.5a)−0.4b+(6b−9.5a)negative 0 point 4 b plus open paren 6 b minus 9 point 5 A close paren (−9.3b+a)+(−7.1b+8.5a)(−9.3b+a)+(−7.1b+8.5a)open paren negative 9 point 3 b plus A close paren plus open paren negative 7 point 1 b plus 8 point 5 A close paren (5.2a+6.1b)+(4.3a+4b)(5.2a+6.1b)+(4.3a+4b)open paren 5 point 2 A plus 6 point 1 b close paren plus open paren 4 point 3 A plus 4 b close paren (4.3a−7.1b)+(5.2a−9.3b)(4.3a−7.1b)+(5.2a−9.3b)open paren 4 point 3 A minus 7 point 1 b close paren plus open paren 5 point 2 A minus 9 point 3 b close paren 9.5a−16.4b

1 answer

To solve the first expression \( 60\left(\frac{1}{4}x - 2\right) + 20\left(\frac{3}{4}y - 4\right) \):

  1. Distribute in both parts: \[ 60\left(\frac{1}{4}x\right) - 60 \cdot 2 + 20\left(\frac{3}{4}y\right) - 20 \cdot 4 \] \[ = 15x - 120 + 15y - 80 \]

  2. Combine like terms: \[ = 15x + 15y - 200 \]

Thus, the equivalent expression is: \[ \boxed{15x + 15y - 200} \]

For the second part, we analyze which of the given expressions are equivalent after simplification.

  1. First expression: \[ -0.4b + (6b - 9.5a) = -0.4b + 6b - 9.5a = (6 - 0.4)b - 9.5a = 5.6b - 9.5a \]

  2. Second expression: \[ (-9.3b + a) + (-7.1b + 8.5a) = -9.3b + a - 7.1b + 8.5a = (-9.3 - 7.1)b + (1 + 8.5)a = -16.4b + 9.5a \]

  3. Third expression: \[ (5.2a + 6.1b) + (4.3a + 4b) = 5.2a + 6.1b + 4.3a + 4b = (5.2 + 4.3)a + (6.1 + 4)b = 9.5a + 10.1b \]

  4. Fourth expression: \[ (4.3a - 7.1b) + (5.2a - 9.3b) = 4.3a - 7.1b + 5.2a - 9.3b = (4.3 + 5.2)a + (-7.1 - 9.3)b = 9.5a - 16.4b \]

  5. Fifth expression: \[ 9.5a - 16.4b \text{ (already simplified)} \]

Therefore, the equivalent expressions are:

  1. \( -9.3b + a + -7.1b + 8.5a \) (simplifies to \( -16.4b + 9.5a \))
  2. \( (4.3a - 7.1b) + (5.2a - 9.3b) \) (also simplifies to \( 9.5a - 16.4b \))
  3. \( 9.5a - 16.4b \) (is equal itself)

So the checked equivalent expressions are:

  1. \( -9.3b + a + -7.1b + 8.5a \)
  2. \( (4.3a - 7.1b) + (5.2a - 9.3b) \)
  3. \( 9.5a - 16.4b \)

Check the three that apply.