Systems of Linear Equations and Word Problems. This section covers: (Note that we solve systems using matrices in the Matrices and Solving Systems with Matrices. В section here.)Introduction to Systems“Systems of equations” just means that we are dealing with more than one equation and variable. В So far, we’ve basically just played around with the equation for a line, which is y = mx + b. But let’s say we have the following situation. В You’re going to the mall with your friends and you have $2. You discover a store that has all jeans for $2. В You really, really want to take home 6 items of clothing because you “need” that many new things. Wouldn’t it be clever to find out how many pairs of jeans and how many dresses you can buy so you use the whole $2. Now, you can always do “guess and check” to see what would work, but you might as well use algebra! В В It’s much better to learn the algebra way, because even though this problem is fairly simple to solve, the algebra way will let you solve any algebra problem – even the really complicated ones. The first trick in problems like this is to figure out what we want to know. В This will help us decide what variables (unknowns) to use. В So what we want to know is how many pairs of jeans we want to buy (let’s say “j”) and how many dresses we want to buy (let’s say “d”). В So always write down what your variables will be: В Let j = the number of jeans you will buy. Let d = the number of dresses you’ll buy. Like we did before, let’s translate word- for- word from math to English. В Always write down what your variables are in the following way: Now we have the 2 equations as shown below. В Notice that the j variable is just like the x variable and the d variable is just like the y. В It’s easier to put in jand d so we can remember what they stand for when we get the answers. This is what we call a system, since we have to solve for more than one. В variable – we have to solve for 2 here. В The cool thing is to solve for 2variables, you typically need 2equations, to solve for 3variables, you need 3equations, and so on. В That’s easy to remember, right? We need to get an answer that works in both equations; this is what we’re doing when we’re solving; this is called solving simultaneous systems, or solving system simultaneously. First we need to set up a system of two equations. The equations will be linear. One of the two will involve the number of people who attended the movie. There are several ways to solve systems; we’ll talk about graphing first. Solving Systems by Graphing. Remember that when you graph a line, you see all the different coordinates (or x/y combinations) that make the equation work. В In systems, you have to make both equations work, so the intersection of the two lines shows the point that fits both equations (assuming the lines do in fact intersect; we’ll talk about that later). В So the points of intersections satisfy both equations simultaneously. В We’ll need to put these equations into they = mx + b (d = mj + b) format, by solving for the d (which is like the y): Now let’s graph: We can see the two graphs intercept at the point (4, 2). В This means that the numbers that work for both equations is 4 pairs of jeans and 2 dresses! We can also use our graphing calculator to solve the systems of equations: (Note that with non- linear equations, there will most likely be more than one intersection; an example of how to get more than one solution via the Graphing Calculator can be found in the. В Exponents and Radicals in Algebra. В section.)Solving Systems with Substitution. Substitution is the favorite way to solve for many students! В It involves exactly what it says: substituting one variable in another equation so that you only have one variable in that equation. В So below are our two equations, and let’s solve for “d” in terms of “j” in the first equation. В Then, let’s substitute what we got for “d” into the next equation. Even though it doesn’t matter which equation you start with, remember to always pick the “easiest” equation first (one that we can easily solve for a variable) to get a variable by itself. So we could buy 4 pairs of jeans and 2 dresses. Note that we could have also solved for “j” first; it really doesn’t matter. В You’ll want to pick the variable that’s most easily solved for. Let’s try another substitution problem that’s a little bit different: Solving Systems with Linear Combination or Elimination. Probably the most useful way to solve systems is using linear combination, or linear elimination. В В The reason it’s most useful is that usually in real life we don’t have one variable in terms of another (in other words, a “y =” situation). The main purpose of the linear combination method is to add or subtract the equations so that one variable is eliminated. В Now let’s see why we can add, subtract, or multiply both sides of equations by the same numbers – let’s use real numbers as shown below. В Remember these are because of the Additive Property of Equality, Subtraction Property of Equality, Multiplicative Property of Equality, and Division Property of Equality: So now if we have a set of 2 equations with 2 unknowns, we can manipulate them by adding, multiplying or subtracting (we usually prefer adding) so that we get one equation with one variable. В For, example, let’s use our previous problem: So we could buy 4 pairs of jeans and 2 dresses. Here’s another example: Types of equations. In the example above, we found one unique solution to the set of equations. This section covers: Introduction to Systems; Solving Systems by Graphing; Solving Systems with Substitution; Solving Systems with Linear Combination or Elimination. Applied Mathematics Department at Brown University. Courses. UNDERGRADUATE COURSES. APMA 0090. Introduction to Modeling Topics of Applied Mathematics, introduced. History. The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by. What is a system of equations? Answer. A system of equation just means 'more than 1 equation.'. A system of linear equations is just more than 1 line, see the picture. В Sometimes, however, there are no solutions (when lines are parallel) or an infinite number of solutions (when the two lines are actually the same line, and one is just a “multiple” of the other) to a set of equations. When there is at least one solution, the equations are consistent equations, since they have a solution. В When there is only one solution, the system is called independent, since they cross at only one point. В When equations have infinite solutions, they are the same equation, are consistent, and are called dependent or coincident (think of one just sitting on top of the other). When equations have no solutions, they are called inconsistent equations, sincewe can never get a solution. В Here are graphs of inconsistent and dependent equations that were created on the graphing calculator: Systems with Three Equations. Let’s get a little more complicated with systems; in real life, we rarely just have two unknowns with two equations. So let’s say at the same store, they also had pairs of shoes for $2. Now we have a new problem: to spend the even $2. Let’s let j = the number of pair of jeans, d = the number of dresses, and s = the number of pairs of shoes we should buy. So far we’ll have the following equations: We’ll need another equation, since for three variables, we need three equations (otherwise, we’d theoretically have infinite ways to solve the problem). В In this type of problem, you would also have/need something like this: В we want twice as many pairs of jeans as pairs of shoes. В Now, since we have the same number of equations as variables, we can potentially get one solution for the system. So, again, now we have three equations and three unknowns (variables). В We’ll learn later how to put these in our calculator to easily solve using matrices (see the. В Matrices and Solving Systems with Matrices section)В , but for now we need to first use two of the equations to eliminate one of the variables, and then use two other equations to eliminate the same variable: Now this gets more difficult to solve, but remember that in “real life”, there are computers to do all this work! The trick to do these problems “by hand” is to keep working on the equations using either substitution or elimination until we get the answers. Remember again, that if we ever get to a point where we end up with something like this, it means there are an infinite number of solutions: В В В 4 = 4. В В В В (variables are gone and a number equals another number and they are the. В same)And if we up with something like this, it means there are no solutions: В В В 5 = 2. В В В (variables are gone and two numbers are left and they don’t equal each other)So let’s go for it and solve В В : So we could buy 6 pairs of jeans, 1 dress, and 3 pairs of shoes. Here’s one more example of a three variable system of equations, where we’ll only use linear elimination: I know – this is really difficult stuff! В But if you do it step- by- step and keep using the equations you need with the right variables, you can do it. В Think of it like a puzzle – you may not know exactly where you’re going, but do what you can in baby steps, and you’ll get there (sort of like life!). And we’ll learn much easier ways to do these types of problems. Also – note that equations with three variables are represented by planes, not lines (you’ll learn about this in Geometry). В They could have 1 solution (if all the planes crossed in only one point), no solution (if say two of them were parallel), or an infinite number of solutions (say if two or three of them crossed in a line). В OK, enough Geometry for now! Algebra Word Problems with Systems. Let’s do more word problems; you’ll notice that many of these are the same type that we did earlier in the Algebra Word Problems. В section, but now we can use more than one variable. В This will actually make the problems easier! Again, when doing these word problems: If you’re wondering what the variable (or unknown) should be when working on a word problem, look at what the problem is asking. В This is usually what your variable is! If you’re not sure how to set up the equations, use regular numbers (simple ones!) and see what you’re doing. В Then put the variables back in! Investment Word Problem: Suppose Lindsay’s mom invests $1. В В The totally yearly investment income (interest) is $2. В How much did Lindsay’s mom invest at each rate?
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