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(2x + 4) + 12

solution



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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

  • multiply factor
    2x + 4 -3x + 2 + x + 3

  • group like terms
    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

2y - 3x = 10

Solution to Example 4

  • We first write the equation in slope intercept form y = mx +b. Put terms in x and constant terms on the right side
    2y = 3x + 10

  • Divide both sides by 2
    y = (3/2)x + 5

  • Now that the equation is in slope intercept form, 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

-3y - 6x = 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

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