# 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

** 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

** 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

** 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

** 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).

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.