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

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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

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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

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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.

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• 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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