Algebra Tutorial

This is a tutorial with detailed solutions and matched exercises on algebra: solve linear equations and equations with absolute value, simplify expressions, find the intercepts of a graph, find the slope of a line and equations of lines. Detailed solutions and explanations ( in red) are provided.

A self test on algebra problems related to topics similar to those in this tutorial can be found in this website.

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 - 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
Interactive Tutorial on Slopes of Lines
Tutorial and Examples on Slopes of Lines
Interactive Tutorial on Lines
Tutorials and Examples on Lines
More Intermediate and College Algebra Questions and Problems with Answers.

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