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# Module 7:Section 4.5

### Systems of Linear Equations and Problem Solving

Question | Answer |
---|---|

Solve: Find two numbers whose sum is 30 and whose difference is 6. | (x + y = 30) + (x - y = 6) Addition Method 2x=36 Divide by 2; x=18plug in 18 and solve for y; 18 + y = 30 The two #s are (18 and 12) CHECK |

Solve: One number is two more than a second number. Twice the first is 4 less than 3 times the second. Find the numbers. | (x=First number and y=second number) Setup: (x=y+2) and 2(x)=3y-4 Substitution Method plug y+2 in for x in second equation and solve for y. (y=8) now plug y into first equation and solve for x. (x=10) Check. |

Solve: Last month Jerry purchased two DVDs and five CDs for $65. This month he bought four DVDs and three CDs for $81. Find the price of each DVD and price of each CD. Set (x=DVD,y=CD) | 2x+5y=65 and 4x+3y=81 (Addition/Elimination) Multiply first equation by -2 to set opposite. -4x-10y=-130 Add to second equation and solve for y. (Y=7) plug 7 in for y in first equation and solve for x. (X=15) Price of DVD $15, price of CD $7 |

Baskets, Inc. is Planning to introduce a new woven basket. The company estimates $500 worth of new equipment will be needed to manufacture this new type of basket and that it will cost $15 per basket to manufacture. Company estimates revenue of $31. | R is revenue and C is cost. R(x)=31x and C(x)=500 + 15x R(x)=C(x) so 31x =500 + 15x Subtract 15 from both sides 16x=500 divide by 16 and round to nearest whole basket. Break even point (x=31) |

Company recently purchased $3000 worth of new equipment to offer new stationary to customers. The cost of producing a package of stationary is $3.00, and it is sold for $5.50. Find the number of packages that must be sold for company to break even. | R(x)=5.50x and C(x)=3000 + 3x set equal. 5.50x=3000+3x Subtract 3 from both sides, 2.50x=3000 Divide by 2.50 Answer is x=1200. Company must sell 1200 packages to break even. |

The measure of the largest angle of the triangle is 40 degrees more than the measure of smallest angle and the measure of the remaining angle is 20 degrees more than the measure of the smallest. Find measure of each angle. | x=smallest angle y=largest z=remaining Triangles =180 degrees so x+y+z=180,y=x+40, z=x+20 Sub y=x+40 & z=x+20 into first equation & solve for x. (x+ (x+40)+(x+20)=180) sub x=40 into 2nd and 3rd equation to get y and z values. z=40+20 y=40+40. (40,80,60) |

The length of a rectangle is 3 feet longer than the width. The perimeter is 30 feet. Find the dimensions of the rectangle. | 3+w=L and 2w+2L=30 Substitue 3+w=L into second equation. 2w+2(3+w)=30 Solve for W. W=6 Plug 6 into first equation and solve. 3+6=L so 9=L |

Sarah and Keith purchased 40 stamps, a mixture of .44 cent and .20cent stamps. Find the number of each type of stamp if they spent $16.40. | x=.44c y=.20c Equations x+y=40 and .44x+.20y=16.40 Mulitply first Eq. by -.44. -.44x-.44y=-17.6 Add to 2nd Eq. solve for y. (y=5) Plug 5 in first equation and solve for x. (x=35) 35 stamps@.44cents 5 stamps@ .20cents |

Puri rowed 18 miles down Delaware River in 2 hours but the return trip took him 4.5 hours, find the rate Puri can row in still waters and find the rate of the current. x= Rate in still water y=rate of current D=r(t) | DownStream (x+y)2=18 upstream (x-y)4.5=18 clear equation of 2 and 4.5 by dividing both sides. x+y=9 ; x-y=4 Add equations solving for x. x=13 plug in 13 for x in first equation and solve for y. y=2.5. Rate in still water 13mph Rate of current 2.5mph |

Samuel has 28 gallons of Saline solution available in two large containers at his pharmacy. One container holds three times as much as the other container. Find capacity of each. | x=large container y=small container x+y=28 and 3y=x Substitue econd equation into x in first equation and solve for y. y=7 Plug 7 in for y in first equation and solve for x. X=21 Large container is 21 gallons of saline and smaller container has 7 gallons |

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