6.1 $$ \begin{array}{l} ext { Simplify the following expressions: } \ ext { 6.1.1 } \quad 2 a-9 b+7 a+3 b \ 6.1 .2 \quad(x-2)^{2}-3 x(2) \ 6.2 \quad ext { Fully factorise the following expressions: } \ ext { 6.2.1 } \quad a^{2}-3 a+2 \ 6.2 .2 \quad 3(2 x-y)+a(y-2 x) \ 6.3 \quad ext { Solve the following equations: } \ ext { 6.3.1 } \quad 3 x+2 x=-75 \ ext { 6.3.2 } \quad 2(x+3)=-12 \ ext { 6.4 mother is now } 38 ext { years older than her son. Ten years ago she was twice as old as } \ ext { her son. How old is her son now? } \ ext { Let the son's age be } x ext { for now. }\end{array} $$ 6

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6.1 $$ \begin{array}{l}	ext { Simplify the following expressions: } \ 	ext { 6.1.1 } \quad 2 a-9 b+7 a+3 b \ 6.1 .2 \quad(x-2)^{2}-3 x(2) \ 6.2 \quad 	ext { Fully factorise the following expressions: } \ 	ext { 6.2.1 } \quad a^{2}-3 a+2 \ 6.2 .2 \quad 3(2 x-y)+a(y-2 x) \ 6.3 \quad 	ext { Solve the following equations: } \ 	ext { 6.3.1 } \quad 3 x+2 x=-75 \ 	ext { 6.3.2 } \quad 2(x+3)=-12 \ 	ext { 6.4 mother is now } 38 	ext { years older than her son. Ten years ago she was twice as old as } \ 	ext { her son. How old is her son now? } \ 	ext { Let the son's age be } x 	ext { for now. }\end{array} $$ 6

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**6.1 Simplify the following expressions:**

*6.1.1*  
We start with  
$$
2a - 9b + 7a + 3b
$$  
Combine like terms:  
$$
2a + 7a = 9a,\quad -9b + 3b = -6b
$$  
Thus, the simplified expression is  
$$
9a - 6b.
$$

*6.1.2*  
We begin with  
$$
(x-2)^2 - 3x(2).
$$  
Recognize that $$3x(2) = 6x$$. First, expand the square:  
$$
(x-2)^2 = x^2 - 4x + 4.
$$  
Substitute back into the expression:  
$$
x^2 - 4x + 4 - 6x.
$$  
Combine like terms:  
$$
-4x - 6x = -10x.
$$  
Thus, the simplified expression is  
$$
x^2 - 10x + 4.
$$

---

**6.2 Fully factorise the following expressions:**

*6.2.1*  
Consider the quadratic  
$$
a^2 - 3a + 2.
$$  
We look for two numbers that multiply to $$2$$ and add to $$-3$$. These numbers are $$-1$$ and $$-2$$. So, we factorise as:  
$$
(a - 1)(a - 2).
$$

*6.2.2*  
We have the expression:  
$$
3(2x-y) + a(y-2x).
$$  
Notice that $$y-2x = -(2x-y)$$. Rewrite $$a(y-2x)$$ as:  
$$
-a(2x-y).
$$  
Now, the expression becomes:  
$$
3(2x-y) - a(2x-y).
$$  
Factor out the common factor $$(2x-y)$$:  
$$
(2x-y)(3-a).
$$

---

**6.3 Solve the following equations:**

*6.3.1*  
The equation is:  
$$
3x + 2x = -75.
$$  
Combine like terms:  
$$
5x = -75.
$$  
Divide both sides by 5:  
$$
x = \frac{-75}{5} = -15.
$$

*6.3.2*  
The equation is:  
$$
2(x+3) = -12.
$$  
Divide both sides by 2:  
$$
x + 3 = -6.
$$  
Subtract 3 from both sides:  
$$
x = -6 - 3 = -9.
$$

---

**6.4 Word Problem:**

"Mother is now 38 years older than her son. Ten years ago she was twice as old as her son. How old is her son now?"

Let the son's current age be $$x$$. Then the mother's current age is  
$$
x + 38.
$$  
Ten years ago, the son's age was $$x - 10$$ and the mother's age was  
$$
(x+38) - 10 = x + 28.
$$  
According to the problem statement, ten years ago the mother's age was twice the son's age:  
$$
x + 28 = 2(x - 10).
$$  
Solve the equation:

1. Expand the right side:  
   $$
   x + 28 = 2x - 20.
   $$
2. Subtract $$x$$ from both sides:  
   $$
   28 = x - 20.
   $$
3. Add 20 to both sides:  
   $$
   x = 48.
   $$

Thus, the son is now $$\boxed{48}$$ years old.
**6.1 Simplify the following expressions:** *6.1.1* $$ 9a - 6b $$ *6.1.2* $$ x^2 - 10x + 4 $$ --- **6.2 Fully factorise the following expressions:** *6.2.1* $$ (a - 1)(a - 2) $$ *6.2.2* $$ (2x - y)(3 - a) $$ --- **6.3 Solve the following equations:** *6.3.1* $$ x = -15 $$ *6.3.2* $$ x = -9 $$ --- **6.4 Word Problem:** The son is now **48** years old.

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