1. Ax + By = c

2.  The Standard Form for a linear equation in two variables, x and y, is usually given as Ax + By = C where, if at all possible, A, B, and C are integers, and A is non-negative, and, A, B, and C have no common factors other than 1.

3.  y = 2x + 3  y – 3 = -2(x + 5)  2x + 3y = 6  1.3x + 2.4y = 9  x – 2y = 8  -6x + 15y = -24  No  Yes  No  Yes  No

4.  This is in slope intercept form.  Subtract 2x from both sides  -2x + y = 3  But, A must be “non-negative” (positive)  Divide all terms by -1  2x – y = -3

5.  This is not in standard form mostly because the x and y terms must both be on the same side with a constant on the other.  Distribute the -2  y – 3 = -2x -10  Add 2x to both sides  2x + y – 3 = -10  Add 3 to both sides  2x – y = -7

6.  This is not in standard form because A, B, and C must be integers (no fractions or decimals).  Multiply each term by a “common denominator”. Since 1.3 is 1 and 3 tenths and 2.4 is 2 and four tenths and 9 would be 9/1, the common denominator is 10.  13x + 24y = 90

7.  This is not in standard form because A is not positive. So, divide each term by -1.  6x – 15y = 24  But, the definition also says they only factor they can have in common is 1. A, B, and C can all be divided by 3.  2x – 5y = 8

8.  Use the intercepts to find the slope.  To find the x intercept, replace y with 0 and solve for x.  2x – (0) = -3, simplify:  2x = -3  Divide both sides by 2.  x = -1.5  (-1.5, 0)

9.  To find the y intercept, replace x with 0 and solve for y.  2(0) – y = -3, simplify  -y = -3  Divide both sides by -1  y = 3  (0, 3)

10.  So, linear equations in any form can be converted to standard form.  As we have seen previously, standard form can be used to find x and y intercepts (that can be used to find slope).

11.  Now use both these points (-1.5, 0) and (0, 3) to find the slope using the slope formula.  (3 – 0)/(0 - -1.5)  3/1.5  3 divided by 1.5 is 2  So, the slope of the line is 2

12.  You could graph this line using the intercepts.  (-1.5, 0) and (0, 3)  Just graph these two points and connect the dots, la, la, la… This point (the x intercept) is called a zero because it is the point at which y is zero. Any x-intercept is a zero or a root.

13.  y – 7 = (2/3)(x - -5)  y – 7 = (2/3)(x + 5)  y – 7 = (2/3)x + 10/3 (or 3 1/3)  Add 7 to both sides  y = (2/3)x + 10 1/3 (or 31/3)  Subtract (2/3)x from both sides  -(2/3)x + y = 31/3  Multiply all terms by the common denominator (3)  -2x + 3y = 31  Divide each term by -1  2x – 3y = -31

14.  Identify:  X intercept  Y intercept  Zero  Slope 3x + 5(0) = 15 3x = 15 x = 5 (5, 0) 3(0) + 5y = 15 5y = 15 y = 3 (0, 3) Remember that the zero is the x intercept. So, it is 5 5

15.  You have $75 to spend on some notebooks that are $3 each and mechanical pencils that are $5 each. Let x be the number of notebooks and y be the number of mechanical pencils.  3x + 5y = 75

16.  What do the intercepts mean in this problem?  First, find each intercept.  The x intercept is found by replacing y with 0.  3x + 5(0) = 75  3x = 75  x = 25  (25, 0)  This is the number of notebooks that can be purchased if no mechanical pencils are purchased.

17.  The y intercept is found by replacing x with 0.  3(0) + 5y = 75  5y = 75  y = 15  (0, 15)  This is the number of mechanical pencils that could be purchased if no notebooks are purchased.

18.  Find the slope of this line and tell its meaning in this problem.  (25, 0) and (0, 15) are the intercepts. Use the slope formula.  (15 – 0)/(0- 25)  15/-25  -3/5  For every three mechanical pencils purchased, 5 notebooks can be purchased.  Continuous or Discrete?  Discrete because these items are counted, not measured.