Example 1: Simplify the expression

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

solution

Example 2: Solve the equation

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

solution

Example 3: If x > -2, simplify the expression

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

solution

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

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

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 - y₁ = m (x - x₁) , where m is the slope and (x₁ , y₁) is any of the two points given above.
Substitute m by its value - 1 and x₁ and y₁ 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

More links and references to pages with algebra problems, tutorials and self tests. Algebra Problems
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