Article: Graphs can be valuable tools in algebra because they allow you to display ideas that you'd usually need numbers for in easy-to-understand pictures. Usually, in beginning algebra, graphing problems are restricted to equations with two variables (usually x and y) and are done on a simple 2-D graph with an x axis and a y axis. With these equations, all you need to do is plug in a value for x, then solve for y (or do the reverse) to get two numbers that correspond to a point on the graph. For example, in the equation y = 3x, if we plug in 2 for x, we get y = 6. This means that the point (2,6) (two spaces to the right of center and six spaces above center) is part of this equation's graph. Equations with the form y = mx + b (where m and b are numbers) are especially common in basic algebra. These equations always have a slope of m and cross the y axis at y = b. What do you do when your equation doesn't use an equals sign? Nothing much different than what you'd normally do, it turns out. For inequalities, which use signs like > ("greater than") and < ("less than"), just solve as normal. You'll be left with an answer that's either less than or greater than your variable. For instance, with the equation 3 > 5x - 2, we would solve just like we would for a normal equation: 3 > 5x - 2 5 > 5x 1 > x, or x < 1. This means that every number less than one works for x. In other words, x can be 0, -1, -2, and so on. If we plug these numbers into the equation for x, we'll always get an answer less than 3. One algebra topic that many beginners struggle with is solving quadratic equations. Quadratics are equations with the form ax2 + bx + c = 0, where a, b, and c are numbers (except that a can't be 0.) These equations are solved with the formula x = [-b +/- (b2 - 4ac)]/2a . Be careful the +/- sign means you need to find the answers for adding and subtracting, so you can have two answers for these types of problems. As an example, let's solve the quadratic formula 3x2 + 2x -1 = 0. x = [-b +/- (b2 - 4ac)]/2a x = [-2 +/- (22 - 4(3)(-1))]/2(3) x = [-2 +/- (4 - (-12))]/6 x = [-2 +/- (16)]/6 x = [-2 +/- 4]/6 x = -1 and 1/3 Solving more than one equation at once may sound super-tricky, but when you're working with simple algebra equations, it's not actually that hard. Often, algebra teachers use a graphing approach for solving these problems. When you're working with a system of two equations, the solutions are the points on a graph that the lines for both equations cross at. For example, let's say we're working with a system that contains the equations y = 3x - 2 and y = -x - 6. If we draw these two lines on a graph, we get one line that goes up at a steep angle, and one that goes down at a mild angle. Since these lines cross at the point (-1,-5), this is a solution to the system. If we want to check our problem, we can do this by plugging our answer into the equations in the system a right answer should "work" for both. y = 3x - 2 -5 = 3(-1) - 2 -5 = -3 - 2 -5 = -5 y = -x - 6 -5 = -(-1) - 6 -5 = 1 - 6 -5 = -5 Both equations "check out," so our answer is right!
Question: What is a summary of what this article is about?
Learn to graph x/y equations. Learn to solve inequalities. Tackle quadratic equations. Experiment with systems of equations.
Article: If your thermostat is set to a high temperature, your gas bill is bound to be expensive. Try lowering your thermostat a few degrees to reduce your bill by 5-10%. For instance, turn the temperature from 70F (21C) to 68F (20C). Don't keep the heat on in your home when it's not necessary. Turn off the heat when you're at work or gone on the weekends. The energy saved will reduce the cost of your gas bill. Some areas in your home need to be warmer than others. For example, it may be important to you to keep the main floor warm, while the basement can be a bit cooler. You can control the heat by installing radiator valves on every radiator in your home. These self-regulating valves allow you to control the temperature of each room by adjusting the amount of hot water goes to the radiator. Personal, or space, heaters use up a lot of energy. Plus, they are an inefficient way of warming up large areas. Avoid using them and instead look for more efficient ways to keep warmlike improving your home's insulation. If you want to warm a small space only, a personal heater may be more effective than heating the whole home with a furnace. Choose a radiant heater, which are more energy-efficient than other models.
Question: What is a summary of what this article is about?
Lower the thermostat. Turn off the heat when you're not home. Save energy with radiator valves. Reduce the use of personal heaters.