Systems of Equations in Two Variables - Chapter 3 Section 1
Typically, problems involving at least two unknowns are solved by translating the problem to two or more equations, such as the slope-intercept form and/or the standard form of the line.
Developing Equations from Sentence Form
When deciphering word problems, we typically try to take a word or phrase which explains what a variable is and represent that word or phrase as a single letter instead. For instance, the energy consumed by your household so far this billing cycle can be represented by the letter E, and the total cost so far in that pay cycle can be represented by the letter C. It is also useful to translate commonly used words into mathematical operations. For example, the product of two things m and c represents multiplication, while the square of c represents squaring the letter c. So the product of the two values of m and the square of c would be $m \times c^2$. I will leave a table of some common and uncommon phrases and their mathematical translations here. Outside of these phrases, it would be useful to find things which would be variables of interest for the problem you're trying to solve. Figure out a single letter for these variables, such as the E for electricity and C for Cost example.
In order to find out how one variable changes with relation to the other variable, combine the variables with the appropriate numbers with the corresponding mathematical operations. For example, suppose I were to say "The cost of your electricity bill increases by the product of your Energy use and the rate of use of 11 cents per kWhr added to the service charge of 12.50". This tells you that the final cost is alone on one side of the equation. This also says that the usage is multiplied by the rate of 11cents per kWhr, $0.11 \times E$. We also know that we should add 12.50 to that product. The final answer for the cost of electricity would be $C=0.11 \times E +12.50$.
This will require plenty of practice to get comfortable with developing equations from information given in sentence form. Just keep in mind to find variables in the problem, find constants, and keep in mind that there are certain words which are equivalent to certain mathematical operations. With that in mind, I am starting something new. In every lecture series post, i am going to link to a homework assignment to allow you to get practice on the material I have covered. For this section, you can find it here.
Identifying Solutions
This section deals with finding the point on a plot of two functions where they meet. This means that a point will be a solution for both lines at the same time. A situation you'll use this concept is when you're shopping around for different cell phone plans which have different amount of data usage and different overage rates. They will cross somewhere; below where they cross, one is the more cost-effective plan while above where they cross, the other is more cost-effective.
In this section, we will be given a point and see whether or not it is the solution of the system of two equations. That is to say, is both equations true at the same time at that point? If yes, then the point is a solution; if no, then the point is not a solution. For example, if we have the pair of equations $f(x)=3x+5$ and $g(x)=6x+2$, we would be given the point (3, 14) and asked to determine if it is a solution to the system of equations. To do this, we would plug the x-value of the point ONLY (3) in to both equations and the if the answer for both equations is the y-coordinate (14). Plugging 3 into f(x), we get $f(3)=3 \times (3)+5=9+5=14$. The equation gives the y-coordinate we were given, so that's half the battle. Now let's try the other equation, $g(3)=6 \times (3)+2=18+2=20$. This gives us a number that is NOT the y-coordinate we were given. Since only one of the equations gives us the appropriate y-value, the point $(3, 14)$ is NOT a solution to the system of equations of $f(x)=3x+5$ and $g(x)=6x+2$.
When we draw two linear functions (lines) on one coordinate system, there are three possible situations. First, they can be parallel lines; in this case, since parallel lines by definition never cross, and since a solution is where the lines cross, there is no solution for parallel lines. Second, two lines are actually the same line; since there is no point on one line which is not also on the other line, meaning that all points on both lines "crosses" the other line, meaning every point on the lines is a solution, which means there are infinitely many solutions. Third, we have a situation where the line are neither parallel nor are they the same line, meaning that they cross only at one point; this means that there is only one point where the lines will be equal, so there is only one solution.
The point here is to determine whether or not the two functions cross, and if they do, find where they cross. If a system has at least one solution, then the system is a consistent system. If a system has no solutions, then it is called inconsistent. So the point can be restated as determining whether or not a system is consistent.
For a practice problem set, go here.
That's it for Chapter this session. If you have any questions, please leave them is the comments. Next chapter will cover how to solve word problems, so subscribe in order to learn this skill. Hopefully, I can do better than most books and professors at explaining how to do this. Like and share this post if you found it helpful. And until next time, stay curious.
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