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Ax + By = c
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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.
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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
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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
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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
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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
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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
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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)
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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)
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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).
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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
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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.
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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
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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
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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
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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.
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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.
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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.
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