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6.4 Standard Form 6.4 – Standard Form REVIEW: Slope-intercept form of a linear equation is y = mx + b 6.4 – Standard Form Standard Form of an equation: Ax + By = C where A, B, & C are REAL #s and A & B are not 0. This is a good form to graph an equation QUICKLY. Standard Form Ax + By = C Rules for Standard Form: No fractions A is not negative (it can be zero, but it CANNOT be negative). By the way, "integer" means no fractions, no decimals. Just clean whole numbers (or their negatives). 6.4 – Standard Form Using Standard Form, you can find the x-intercept and y-intercept, and then graph the equation. To find the intercepts, you substitute 0 for both the x and y. 6.4 – Standard Form Example: Find the x and y intercepts for 3x + 4y = 24. To find the x intercept, substitute 0 for the y. 3x + 4(0) = 24 3x = 24 x=8 So, when y = 0, x = 8. Your x-int. is (8, 0). 6.4 – Standard Form To find the y-int. substitute 0 for the x 3(0) + 4y = 24 4y = 24 y=6 So your y-int is (0, 6) Using your x and y-int, you can now graph the equation. 6.4 – Standard Form y 5 4 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 1 2 3 4 5 x 6.4 – Standard Form Example: Graph the equation 4x + 6y = 2 y 5 4 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 1 2 3 4 5 x 6.4 – Standard Form Using Standard Form, you can write equations for vertical and horizontal lines. (You can’t write vertical lines in slope-intercept form) 6.4 – Standard Form Example: Graph the following y= -2 x=4 When graphing these, say draw a line through the ___ - axis at the number. y –5 –4 –3 –2 y 5 5 4 4 3 3 2 2 1 1 –1 –1 1 2 3 4 5 x –5 –4 –3 –2 –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 1 2 3 4 5 x 6.4 – Standard Form If we are looking for the intercepts of an equation, standard from is the easiest to use. Therefore, we may want to change slope-intercept equations to standard form. 6.4 – Standard Form Example: Write y 2 x 6 in Standard Form. 3 First we must move the x to the other 2 side. So we add x to both sides to 3 get: 2 x y 6 3 6.4 – Standard Form Now we must make the numbers whole. So we must multiply by 3 to get rid of the fraction. 2 3 x y 6 3 2 x 3 y 18 Ex. 2: Write 3 y 4 (in x standard 2) form. 4 3 y 4 ( x 2) 4 Given 4(y + 4) = 3(x – 2) Multiply by 4 to get rid of the fraction. 4y + 16 = 3x – 6) Distributive property 4y = 3x – 22 Subtract 16 from both sides 4y – 3x= – 22 Subtract 3x from both sides – 3x + 4y = – 22 Format x before y 3x – 4y = 22 Multiply by -1 in order to get a positive coefficient for x. Ex. 3: Write the standard form of an equation of the line passing through (5, 4), -2/3 2 y 4 ( x 5) 3 Given 3(y - 4) = -2(x – 5) Multiply by 3 to get rid of the fraction. 3y – 12 = -2x +10 Distributive property 3y = -2x +22 Add 12 to both sides 3y + 2x= 22 2x + 3y = 22 Add 2x to both sides Format x before y Ex. 4: Write the standard form of an equation of the line passing through (-6, -3), -1/2 1 y 3 ( x 6) 2 Given 2(y +3) = -1(x +6) Multiply by 2 to get rid of the fraction 2y + 6 = -1x – 6 Distributive property 2y = -1x – 12 Subtract 6 from both sides 2y + 1x= -12 x + 2y = -12 Subtract 1x from both sides Format x before y Ex. 6: Write the standard form of an equation of the line passing through (5, 4), (6, 3) y2 y1 m x2 x1 First find slope of the line. 3 4 1 m 1 65 1 y 4 1( x 5) y – 4 = -1x + 5 y = -1x + 9 y+x=9 x+y=9 Substitute values and solve for m. Put into point-slope form for conversion into Standard Form Ax + By = C Distributive property Add 4 to both sides. Add 1x to both sides Standard form requires x come before y. Ex. 7: Write the standard form of an equation of the line passing through (-5, 1), (6, -2) y2 y1 m x2 x1 m First find slope of the line. 2 1 3 3 6 (5) 6 5 11 Substitute values and solve for m. y 1 3 ( x 5) 11 Put into point-slope form for conversion into Standard Form Ax + By = C – 1) = -3(x + 5) 11y – 11 = -3x – 15 11(y 11y 11y = -3x – 4 Multiply by 11 to get rid of fraction Distributive property Add 4 to both sides. + 3x = -4 3x + 11y = -4 Add 1x to both sides Standard form requires x come before y. 6.4 – Standard Form Writing equations in the REAL WORLD: You are working two jobs during the summer. You are mowing lawns and delivering newspapers. You make $12/hour mowing lawns and $5/hour delivering newspapers. If you made a total of $130, write an equation in standard form. 12x + 5y = 130 6.4 – Standard Form Example #2: You are training to participate in the annual Ironman Championship in Kona, Hawaii. You need to burn a total of 500 calories per day to get in proper shape. Write an equation in standard form to find the minutes you would need to workout each day if you were to just swim and run. 6.4 – Standard Form Activity Calories burned per minute Bicycling 10 Running 11 Hiking 7 Swimming 12 Walking 2 Rowing 10