دو معادله، دو مجهول 1فصل: بخش ریاضی / درس: جبر، معادلات و نابرابری ها / درس 14
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Two Equations, Two Unknowns - I
Two equations with two variables. So far in the study of algebraic equations we have looked at solving single equations with only one variable. For example, something like 2x plus 7 equals 15. What happens if there is more than one variable in an equation? Suppose we had something like 2x plus 3y equals 15.
Now what would it mean for someone to come and tell us, solve this equation? How would we find values that work in this equation? Well, certainly one possible value if x equals zero then y could equals five. So that would be a solution. Others would be if x equals 3 and y equals 3, x equals 6 and y equals 1. Those are also values that make it work.
Of course, there’s no restriction that either variable has to be positive, so other solutions include, x equals 9 y equals negative 1 or x equals negative 3 and y equals positive 7. And as you can imagine, we could make x more, more and more negative and make y more and more positive, or vice versa. So we can get quite a few solutions of that sort.
Also, there’s no restrictions that the variables must be integers, so other solutions include things like x equals 7 and a half, y equals 0. Or x equals 4 and y equals 2 and a third. So, just on this page, notice we have one, two, three, four, five, six, seven solutions for this. And it’s certainly clear we could get many, many more.
In fact, one equation with two variables typically would have an infinite number of solutions. Notice all those solutions, if plotted on an x, y graph, would lie on a straight line. So all, the seven solutions that we mention, those are the seven dots on this diagram, and they all lie in a straight line.
Now, for reasons we will discuss later, in the Coordinate Geometry module, any single equation with just x and y, neither variable raised to a power or in a fraction, can be represented by a line in the x y plane. So right now you don’t need to worry about graphing those. You don’t need to worry about how would you find the slope of the line or any of that.
All you need to do is just have that idea, just that association that an equation with x and y is represented by a line. That’s all you need to know for this discussion here. So the first big idea is, no one can ask you to solve a single equation with two variables, because it would have an infinite number of solutions. A line passes through an infinite number of points, and every single one of those points is a solution.
So there’s no way anyone could legitimately ask you to solve because they’d be asking you to solve for an infinity of things all at once. Now suppose we have two equations, each with two variables. This is called a system of equation. The values of x and y must satisfy both equations simultaneously. Well, this is interesting.
If each equation is a line. Then it makes sense that the unique point where those two lines intersect would be the single point that satisfies both equations. So if you pick one random line, and pick another random line, chances are very good that they’re going to intersect some where. And they intersect at one point, and that one point would be the solution.
So algebraically, when we’re finding that solution, what we’re doing is we’re finding the point geometrically where they intersect. So big idea number two is if we have a system of two equations with two unknowns we generally can solve for unique values of x and y. How do we solve a system of equations for these values? There are two strategies.
One is substitution and the other is called either elimination, some sources also call it linear combination. I’ll be calling them substitution and elimination. The goal of both of these methods is to reduce the two equations to a known situation, to a one equation one unknown situation, which is one in which we, we already know how to find the solution.
So what we’re doing, and this is often true of mathematics, we’re turning a problem we don’t know how to solve into a problem we do know how to solve. That’s very typical for mathematics. So, the substitution method. In this method, we will first solve one equation, either one, for one of the variables.
In this equation, we’ll get one variable by itself on one side of the equation. So those two equations I gave a moment ago. One of the equations was x plus 2y equals 11, and that’s an equation where it’s particularly easy to get x by itself. All I’m going to do is subtract 2y from both sides, and I get x equals 11 minus 2y.
So hold on to that for a second. x equals 11 minus 2y. Now let’s look at the other equation. We can replace the x in the other equation with the expression that x equals because x equals 11 minus 2y means that where ever there is an x, we can remove the x and replace it with the thing that it equals.
So here’s the other equation, and we’re just going to write the same equation again but we’re going to replace that x with 11 minus 2y. Well, now we have a single equation with y. So now we just use our ordinary solving. We’ll distribute. We’ll combine the the like terns.
We’ll subtract the 22 from both sides, we get negative y equals negative 7 multiplied by negative 1 we get y equals 7. So now we have solved for one of the two values, we solved for y. We still have to solve for x. Now we plug this value of y back into the equation that was solved for x. So we had x equals 11 minus 2y.
Well now we know that y equals 7. So we’ll just plug that in. 11 minus 14 is negative 3, and so that point, x equals negative 3, y equals positive 7, that is the solution. Notice that the substitution method is most useful when, in one of the two equations, the coefficient of one of the variables equals positive one or negative one.
If all coefficients in, of x and y in the two equations are unequal to positive or negative one, then solving for any variable will create fractions, which makes the solution more cumbersome. So for example, suppose we have this as our system. Suppose we try to solve the first equation for x. Okay, well we can subtract by y from both sides, then we divide by 4.
All right, well immediately get into fractions. So this would not be fun to substitute. Yes, mathematically we could solve the equation this way. We’d have to wade through a bunch of factions. But we prefer not to have to do it. In systems in which substitution is not convenient, we would use elimination.
We will cover the elimination method in the next lesson. In summary, a system of equations, two equations with two variables, typically has a single unique solution. And again, this would be where the two lines are intersecting. That’s the point that we’re finding. We can solve, with either substitution or elimination.
Substitution works best, when one of the variables has a coefficient of plus or minus 1. And again, in the next lesson, we’ll talk about elimination.
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