Subtract the following binomials. (Both by vertical and horizontal arrangement) $$ \begin{array}{ll} ext { a) } 2 x+3 y ext { from } 3 x+5 y & ext { b) } 5 x-4 y ext { from } 2 x+5 y \ ext { c) } 8 x y+2 x ext { from } 5 x y-2 x & ext { d) } 4 x-5 z ext { from } 6 x-4 z \ ext { e) } 15 x^{2} y^{3}-3 x y^{2} ext { from }-5 x^{2} y^{3}-3 x y^{2} & ext { d) } 5+2 m n ext { from } 6-4 ext { mn }\end{array} $$

Question

Subtract the following binomials. (Both by vertical and horizontal arrangement) $$ \begin{array}{ll}	ext { a) } 2 x+3 y 	ext { from } 3 x+5 y & 	ext { b) } 5 x-4 y 	ext { from } 2 x+5 y \ 	ext { c) } 8 x y+2 x 	ext { from } 5 x y-2 x & 	ext { d) } 4 x-5 z 	ext { from } 6 x-4 z \ 	ext { e) } 15 x^{2} y^{3}-3 x y^{2} 	ext { from }-5 x^{2} y^{3}-3 x y^{2} & 	ext { d) } 5+2 m n 	ext { from } 6-4 	ext { mn }\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's subtract each pair of binomials both vertically and horizontally.

### a) Subtract **$$2x + 3y$$** from **$$3x + 5y$$**

**Vertical Arrangement:**
```
   3x + 5y
-  2x + 3y
-----------
     x + 2y
```

**Horizontal Arrangement:**
$$
(3x + 5y) - (2x + 3y) = 3x - 2x + 5y - 3y = x + 2y
$$

---

### b) Subtract **$$5x - 4y$$** from **$$2x + 5y$$**

**Vertical Arrangement:**
```
   2x + 5y
-  5x - 4y
-----------
  -3x + 9y
```

**Horizontal Arrangement:**
$$
(2x + 5y) - (5x - 4y) = 2x - 5x + 5y + 4y = -3x + 9y
$$

---

### c) Subtract **$$8xy + 2x$$** from **$$5xy - 2x$$**

**Vertical Arrangement:**
```
    5xy - 2x
-  8xy + 2x
------------
  -3xy - 4x
```

**Horizontal Arrangement:**
$$
(5xy - 2x) - (8xy + 2x) = 5xy - 8xy - 2x - 2x = -3xy - 4x
$$

---

### d) Subtract **$$4x - 5z$$** from **$$6x - 4z$$**

**Vertical Arrangement:**
```
   6x - 4z
-  4x - 5z
----------
    2x + z
```

**Horizontal Arrangement:**
$$
(6x - 4z) - (4x - 5z) = 6x - 4x - 4z + 5z = 2x + z
$$

---

### e) Subtract **$$15x^2 y^3 - 3xy^2$$** from **$$-5x^2 y^3 - 3xy^2$$**

**Vertical Arrangement:**
```
   -5x²y³ - 3xy²
- 15x²y³ - 3xy²
---------------
  -20x²y³ + 0xy²
```
*Since $$0xy²$$ adds no value, the simplified result is:*

$$
-20x^2 y^3
$$

**Horizontal Arrangement:**
$$
(-5x^2 y^3 - 3xy^2) - (15x^2 y^3 - 3xy^2) = -5x^2 y^3 - 15x^2 y^3 - 3xy^2 + 3xy^2 = -20x^2 y^3
$$

---

### f) Subtract **$$5 + 2mn$$** from **$$6 - 4mn$$**

*(Note: This was labeled as "d)" in your list, but we'll label it as "f)" to maintain order.)*

**Vertical Arrangement:**
```
    6 - 4mn
-   5 + 2mn
-----------
    1 - 6mn
```

**Horizontal Arrangement:**
$$
(6 - 4mn) - (5 + 2mn) = 6 - 5 - 4mn - 2mn = 1 - 6mn
$$

---

### Summary of Results:

- **a)** $$x + 2y$$
- **b)** $$-3x + 9y$$
- **c)** $$-3xy - 4x$$
- **d)** $$2x + z$$
- **e)** $$-20x^2 y^3$$
- **f)** $$1 - 6mn$$

Feel free to ask if you need further clarification on any of these subtractions!
Here are the results of subtracting the given binomials: a) $$x + 2y$$ b) $$-3x + 9y$$ c) $$-3xy - 4x$$ d) $$2x + z$$ e) $$-20x^2 y^3$$ f) $$1 - 6mn$$

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