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More Linear Equations

More Linear Equations. Cramer’s rule: derived by solving the general system by elimination. Uses the determinant notation. a x + b y = e c x + d y = f a c x - b c y = -c e a c x + a d y = a f ady – bcy = af – ce (ad – bc) y = af – ce af – ce

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More Linear Equations

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  1. More Linear Equations Cramer’s rule: derived by solving the general system by elimination. Uses the determinant notation.

  2. a x + b y = e • c x + d y = f • a c x - b c y = -c e • a c x + a d y = a f • ady – bcy = af – ce • (ad – bc) y = af – ce • af – ce • y = ---------- • ad – bc • adx + bdy = de • -bcx - bdy = -bf • adx – bcx = de – bf • (ad – bc)x = de – bf • de – bf • x = ------------ • ad – bc Cramer introduced the determinant notation | x y | | | = x w – z y | z w| for a difference of products. It make it easier to remember what order to subtract. Cramer’s rule for 2 by 2 system of linears:

  3. Example: 3 x – 4 y = 2 7 x + 5 y = 13 Another example:

  4. Cramer’s Rule doesn’t work when the determinant = 0 Example: 2x + 3y = 0 6x + 9y = 1 Another example: 2x + 3y = 1 6x + 9y = 3

  5. Lots of questions can be answered by setting up and solving a system of two equations in two variables. Common #

  6. Almost always there is more than one way to set up the equations. Example: An unknown number is multiplied by 7 and added to 5. The resulting number is divided into 40, and 3 is obtained. What is the unknown number? One setup: Let x be the desired number. Then 40/(7*x + 5) = 3 Solve for x. Another setup. Let x be the unknown number and y be the resulting number. Then y = 7*x + 5 and 40/y = 3. Solve for y, then for x.

  7. Common #4 A rectangular blue box of an unknown height has a width of 3 in. A red box with the same height is 12 inches wider. The area of the red box is 5 sq in. more than three times the area of the blue box. Then the heights of the two boxes must be _________in.

  8. Like common #10 A mix of peanuts and walnuts is to be made up and sold for $2.25 per pound. Peanuts cost $1.00 per lb and walnuts cost $2.00 per lb. If the cost of the mix is to be $1.75 per lb., how many lbs of peanuts must there be in 10 lbs of mix?

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