Solutions to Algebra Problems

Detailed solutions to algebra problems are presented.

Solution to Problem 1:

• Given the equation
  
  5(-3x - 2) - (x - 3) = -4(4x + 5) + 13
  
  
• Multiply factors.
  
  -15x - 10 - x + 3 = -16x - 20 +13
  
  
• Group like terms.
  
  -16x - 7 = -16x - 7
  
  
• Add 16x + 7 to both sides and write the equation as follows
  
  0 = 0
  
  
• The above statement is true for all values of x and therefore all real numbers are solutions to the given equation.

Solution to Problem 2:

• Given the algebraic expression
  
  2(a -3) + 4b - 2(a -b -3) + 5
  
  
• Multiply factors.
  
  = 2a - 6 + 4b -2a + 2b + 6 + 5
  
  
• Group like terms.
  
  = 6b + 5

Solution to Problem 3:

• Given the expression
  
  |x - 2| - 4|-6|
  
  
• If x < ;2 then x - 2 < 2 and if x - 2 < 2 the |x - 2| = -(x - 2).
  
  
• Substitute |x - 2| by -(x - 2) and |-6| by 6 .
  
  |x - 2| - 4|-6| = -(x - 2) -4(6) = -x -22

Solution to Problem 4:

• The distance d between points (-4 , -5) and (-1 , -1) is given by
  
  d = sqrt[ (-1 - -4) ² + (-1 - -5) ² ]
  
  
• Simplify.
  
  d = sqrt(9 + 16) = 5

Solution to Problem 5:

• Given the equation
  
  2x - 4y = 9
  
  
• To find the x intercept we set y = 0 and solve for x.
  
  2x - 0 = 9
  
  
• Solve for x.
  
  x = 9 / 2
  
  
• The x intercept is at the point (9/2 , 0).

Solution to Problem 6:

• Given the function
  
  f(x) = 6x + 1
  
  
• f(2) - f(1) is given by.
  
  f(2) - f(1) = (6*2 + 1) - (6*1 + 1) = 6

Solution to Problem 7:

• Given the points (-1, -1) and (2 , 2), the slope m is given by
  
  m = (y2 - y1) / (x2 - x1) = (2 - -1) / (2 - -1) = 1

Solution to Problem 8:

• Given the line
  
  5x - 5y = 7
  
  
• Rewrite the equation in slope intercept form y = mx + b and identify the value of m the slope.
  
  -5y = -5x + 7
  
  y = x - 7/5
  
  
• The slope is given by the coefficient of x which is 1.

Solution to Problem 9:

• To find the equation of the line through the points (-1 , -1) and (-1 , 2), we first use the slope m.
  
  m = (y2 - y1) / (x2 - x1) = (2 - -1) / (-1 - -1) = 3 / 0
  
  
• The slope is indefined which means the line is perpendicular to the x axis and its equation has the form x = constant. Since both points have equal x coordinates -1, the equation is given by:
  
  x = -1

Solution to Problem 10:

• The equation to solve is given by.
  
  |-2x + 2| -3 = -3
  
  
• Add 3 to both sides of the equation and simplify.
  
  |-2x + 2| = 0
  
  
• |-2x + 2| is equal to 0 if -2x + 2 = 0. Solve for x to obtain
  
  x = 1

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