Problem: Article: If a linear equation has one integral solution, then it must have infinitely many integral solutions. Here is a brief algebraic statement of the proof: Ax+By=C{\displaystyle Ax+By=C} A(x+B)+B(yA)=C{\displaystyle A(x+B)+B(y-A)=C} .. (Adding a B to x while subtracting A from y results in the same solution.) The pattern of infinite solutions begins with the single solution that you identified. In this case, your solution is the coordinate pair (x,y)=(75,102){\displaystyle (x,y)=(-75,-102)}. To find a new solution for x, add the value of the coefficient of y. In this problem, beginning with the solution x=-75, add the y coefficient of -64, as follows: x=75+(64)=139{\displaystyle x=-75+(-64)=-139} Thus, a new solution for the original equation will have the x value of -139. To make the equation remain balanced, when you add to the x term, you must then subtract from the y term. For this problem, beginning with the solution y=-102, subtract the x coefficient of 87, as follows: y=10287=189{\displaystyle y=-102-87=-189} Thus, a new solution for the original equation will have the y coordinate of -189. The new ordered pair should be (x,y)=(139,189){\displaystyle (x,y)=(-139,-189)}. To verify that your new ordered pair is a solution to the equation, insert the values into the equation and see if it works. 87x64y=3{\displaystyle 87x-64y=3} 87(139)64(189)=3{\displaystyle 87(-139)-64(-189)=3} 3=3{\displaystyle 3=3} Because the statement is true, the solution works. The values for x will fit a pattern of the original solution, plus any multiple of the B coefficient. You can write this algebraically as follows: x(k)=x+k(B), where x(k) represents the series of all x solutions, and x is the original x value that you solved. For this problem, you can say: x(k)=7564k{\displaystyle x(k)=-75-64k} y(k)=y-k(A), where y(k) represents the series of all y solutions, and y is the original y value that you solved. For this problem, you can say: y(k)=10287k{\displaystyle y(k)=-102-87k}
Summary: Recognize that infinitely many solutions exist. Identify your original solution values for x and y. Add the y-coefficient B to the x solution. Subtract the x-coefficient A from the y solution. Check the solution. Write a general solution.

Problem: Article: Electricity is often a large monthly expense. Do both your wallet and the environment a favor and cut back on your home's energy use. Seal cracks in your home to better insulate it and reduce the need for heating and air conditioning. Set your thermostat to a higher temperature in the summer and a lower one in the winter. Unplug appliances when not in use and remember to turn off the lights. Change your computer's settings so that it enters a very low-consumption "hibernate" mode when you're not using it. When buying new appliances, go with ones that have energy-saving features. Shop around for a different service provider for your insurance, phone, and Internet. There may be new, better offers from carriers that have been introduced since you first started your plan. Evaluate whether your current service or coverage level is still right for you. Also, try to negotiate with your current provider for a lower rate. If you mention your intention to switch to a competitor, they may be more likely to give you a better offer. If you are looking to buy a new vehicle, make sure your purchase has high value. Buy a model known to last a long time with low maintenance costs. You will also immediately begin to save money with a car that has better gas mileage, especially if you commute via car to work. If your credit rating has improved since buying a home, it may make sense to refinance your mortgage. Because many homeowners find their credit getting better over time, they may qualify for a lower interest rate than they did a few years earlier. Refinancing can result in lower monthly payments and/or less money going towards interest. Consult your mortgage company to figure out if refinancing is right for you.
Summary: Reduce your energy consumption. Consider reducing service levels. Buy a reliable car with good gas mileage. Refinance your mortgage.

Problem: Article: This can help you monitor for new mole growth. You also should watch for discoloration or growth of old moles. If you have more than 100 moles, you are a higher risk for skin cancer. You should make an appointment with a dermatologist. Before you even consider removing a mole, you should know the type of mole and its symptoms. Some moles can be safe to remove, while others are not. Atypical mole Atypical moles, or dysplastic moles, can look suspicious in color and size. They can sometime be larger than an eraser head, have an odd shape, or have many colors. If you have this type of mole, consult your doctor to be sure it is not cancerous. Congenital mole This is the type of mole you are born with. Roughly, 1 in 100 people are born with moles. They can range in size from tiny (size of a pin head) to giant (larger than a pencil eraser). Doctors suspect that people born with giant moles are at a higher risk for skin cancer. Spitz nevus This mole looks pink, raised, and dome shaped. It often looks like melanoma. It may ooze. It may itch or bleed. These are uncommon and mostly benign. Acquired Mole This term means moles that are acquired after birth. These are often called common moles. A common way to do so is to remember the "ABCD" rule. If you suspect that you have melanoma, seek medical attention immediately. Asymmetry The mole looks uneven, or one side does not match the other in size, shape, or color. Border that is irregular The mole has ragged, blurred, or uneven edges. Color that is uneven The mole has multiple shades, including black, brown, tan, or blue. Diameter The mole is large in diameter (typically around inch wide). Evolving The mole changes size, shape, and/or color in weeks or months time.
Summary:
Perform a self-skin check. Count your moles. Recognize different mole types. Identify melanoma symptoms.