__Example 1:__ Simplify the expression

2(-4a - 5b) - (8 + b) + b + (-2b + 4) - 5a
__Solution to Example1__

given

2(-4a - 5b) - (8 + b) + b + (-2b + 4) - 5a

multiply factors

-8a - 10b - 8 - b + b -2b + 4 - 5a

group like terms

- 13a - 12b - 4

__Matched Exercise 1__ Simplify the expression

2(a - 8b) - (5 - b) + b + (6b - 9) - a

solution

__Example 2:__ Solve the equation

2(-3x - 5) - (8 - x) = -2(2x + 4) + 12
__Solution to Example 2__

given

2(-3x - 5) - (8 - x) = -2(2x + 4) + 12

multiply factors

-6x -10 - 8 + x = -4x - 8 + 12

group like terms

-5x - 18 = -4x + 4

add 18 to both sides

-5x -18 + 18 = -4x + 4 + 18

group like terms

-5x = -4x + 22

add 4x to both sides

-5x + 4x = -4x + 22 +4x

group like terms

-x = 22

multiply both sides by -1

x = -22

Check the solution

left side:2(-3*(-22) - 5) - (8 - (-22)) = 92

right side:-2(2(-22) +4) + 12 = 92

Conclusion

x = -22 is the solution to the given equation

__Matched Exercise 2:__ Solve the equation

2(-x - 5) - (- 6 + x) = -3(2 x + 4) + 12
solution

__Example 3:__ If x > -2, simplify the expression

2| x + 2 | - 3x - (-2 - x) + | 6 - 9 |
__Solution to Example 3__

To simplify the given expression, we need to simplify the terms with absolute value using definition of absolute value.

if x > = 0 , | x | = x

if x < 0 , | x | = -x

According to the definition of the absolute value above,

x > - 2 (given above) is equivalent to x + 2 > 0

if x + 2 > 0 then | x + 2 | = x + 2

the above definition gives

| 6 - 9 | = | - 3 | = 3

the whole expression given above can now
be written as

2(x + 2) - 3x - (-2 - x) + 3

expand product

2x + 4 -3x + 2 + x + 3

group like terms and simplify

(2x - 3 x + x) + (4 + 2 + 3) = 9

**Matched Exercise 3**__:__ If x
> 3, simplify the expression

2| x - 3 | + 6x - (2 - 3x) + | 9 - 20 |
solution

__Example 4:__ Find the slope and the y-intercept of the line given by the equation

2 y - 3 x = 10
__Solution to Example 4__

We first write the equation in slope
intercept form y = m x +b. Put terms in x and constant terms on the right side

2 y = 3 x + 10

Divide both sides by 2

y = (3/2)x + 5

Now that the equation is in slope intercept form y = m x + b, we identify the slope as the coefficient of x and is equal to 3/2 and the y intercept as (0 , 5).

**Matched Exercise 4**__:__ Find
the slope and the y-intercept of the line given by the equation

-3 y - 6 x = 7
solution

__Example 5:__ Find the equation of the line passing through the points (2 , 3) and (4 , 1).

__Solution to Example 5__

We first calculate the slope m

m = (1 - 3) / (4 - 2) = -1

We now use the point-slope form of a line to find the equation of the line

y - y_{1} = m (x - x_{1}) , where m is the slope and (x_{1} , y_{1}) is any of the two points given above.

Substitute m by its value - 1 and x_{1} and y_{1} by 2 and 3 respectively, we obtain the equation of the line.

y - 3 = - 1(x - 2)

in slope intercept form the equation is written as

y = - x + 5

**Matched Exercise 5**__:__ Find
the equation of the line passing through the points (0 , 3) and (-1 , 1).

solution

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