Elementary Algebra Solutions

1. Evaluate  2x² - y² - z  if  x = 3,  y = -1  and  z = -2
   Substitute the values for the variables; follow order of operations to evaluate:
      2(3)² - (-1)² - (-2)
   = 2(9) - (1) - (-2)
   = 18 - (1) + (+2)
   = 17 + 2
   = 19
    
2. Solve for x:   1 - 3(x - 4) = 2(3x + 1) - 7
   Distribute:
   1 - 3x +12 = 6x + 2 - 7
   Collect like terms:
   -3x +13 = 6x - 5
   Add 3x to both sides of the equal sign:
   -3x +13 + 3x = 6x - 5 + 3x
   13 = 9x - 5
   Add 5 to both sides:
   18 = 9x
   Divide both sides by 9:
   2 = x
    
3. Solve for y:   ax + by = c
   Subtract ax from both sides:
   by = c - ax
   Divide both sides by b:
   
    
4. Draw a picture of a rectangle.
   Setup:   Let W = width; L = Length
   Translate the phrases into math:
   The length of a rectangle is 8 feet longer than its width:  L = W + 8
   The perimeter is the total distance around the rectangle:
   236 = W + L + W + L
   236 = 2L + 2W
   Substitute L = W + 8 and solve:
   236 = 2(W + 8) + 2W
   Distribute the 2:
   236 = 2W + 16 + 2W
   Collect like terms:
   236 = 4W + 16
   Subtract 16 from both sides:
   220 = 4W
   Divide both sides by 4:
   55 = W
   Be sure to answer the question:  'What is the length of the rectangle?'
   L = W + 8 = 55 + 8 = 63
   The length of the rectangle is 63 feet 
    
5. The graph of  2x + y = 4  is
   
   Find two points that satisfy the given equation; plot them, draw a line through the points.
   Easy points to find are the intercepts:

   x-intercept:	2x + (0) = 4 2x = 4 x = 2
   y-intercept:	2(0) + y  = 4 y = 4

   The two points are:  (2, 0) and (0, 4).  Plot them and draw the line.
    
6. Setup:  Let J = Jane's annual salary; S = Sam's annual salary
    
   Translate the words into math:
   Jane donates 2% of her annual salary:  0.02J
   Sam donates 1% of his annual salary:   0.01S
    
   The total of Jane and Sam's donations is $620:   0.02J + 0.01S = 620
   Jane makes $1000 more than Sam:  J = S + 1000    or    J - 1000 = S
    
   Substitute  J - 1000 for S in the first equation:
   0.02J + 0.01(J - 1000) = 620
   Solve for J:
   0.02J + 0.01J - 10 = 620
   0.03J - 10 = 620
   Add 10 to both sides:
   0.03J = 630
   Divide both sides by 0.03:
   J = 21000
   Jane's annual salary is $21,000
    
7. Use FOIL to multiply:
       (2a - 3b)(a + 2b)
   = 2a² + 4ab - 3ab - 6b²
   = 2a² + ab - 6b²
    
8. Factor the denominator and cancel common factors:
   
    
9. Raise each factor inside the parentheses to the second power and mulitiply:
   
    
10. Solve for x:    x² - 4a² = 0
    This is a quadratic equation with two solutions.
    Isolate x²:
     x² = 4a²
    Solve for x¹ by taking the positive and negative square root of the right side of the equal sign:
    
    x = ± 2a

On to the next review:  College Level Algebra