How to Solve Two Equations with Two Unknowns – I

Could you use a refresher on how to solve two equations with two unknowns? If so, or if you’ve never been introduced to this concept, I invite you to watch this video, and the next video, which is Part Two in this series.

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Transcript: How to Solve Two Equations with Two Unknowns

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 + 7 = 15.

What happens if there is more than one variable in an equation? Suppose we had something like 2x + 3y = 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=0, then y could equal 5, so that would be a solution. Others would be, if x = 3 and y = 3, x = 6 and y = 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 = 9, y = -1), or ( x = -3 and y = +7). And as you can imagine, we could make x more and more negative and make y more and more positive–or vice versa. So we could get quite a few solutions of that sort. Also there is no restriction that the variables must be integers so other solutions include things like x = 7 1/2 y = 0 or x = 4 and y = 2 1/3.

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.

Reading the Diagram

So the seven solutions that we mentioned, those are the seven dots on this diagram. And they all lie on 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 will 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.

Big Idea Number One

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 equations. 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, will be the single point that satisfies both equations. So you pick one random line and pick another random line, chances are very good that they are going to intersect somewhere, and they intersect at one point and that one point would be the solution.

Big Idea Number Two

Algebraically, when we’re finding that solution, what we’re doing is we’re finding the point geometrically where they intersect. So Big Idea #2 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 will 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-two-unknown situations to a one-equation-one-unknown situation which is one in which we already know how to find the solution.

The Substitution Method

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 there’s two equations I gave a moment ago, one of the equations was x + 2y = 11. And that’s an equation where it’s particularly easy to get x by itself. What I am going to do is subtract 2y from both sides and I get x=11-2y. So hold onto that for a second, x=11-2y. Now, lets look at the other equation.

We can replace the x in the other equation with the expression that x equals. This is because x=11-2y means that where ever there is an x, we can remove the x and replace it by the thing that it equals. So, here’s the other equation, and we’re just gonna write the same equation again but we’re gonna replace that x with 11 minus 2y. Where now we have a single equation with y.

So now we just use our ordinary solving, we’ll distribute, combine the

like terms, we will subtract the 22 from both sides, we get -y= -7 multiply by -1 we get y=7. So now we have solved for one of the two values, we have solved for y, we still have to solve for x. Now we plug this value go y back into the equation that was solved for x.

So we had x = 11- 2y, well now we know that y = 7. So we’ll just plug that in, 11- 14 is -3. And so that point x equals -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 coefficients of one of the variables equals positive 1 or negative 1.

Fractions and the Elimination Method

If all coefficients 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 supposed we have this as our system. Supposed we try to solve the first equation for x. Okay where we can spread 5y on both sides then we divide by 4.

Immediately we get into fractions. This would not be fun to substitute. Yes, mathematically we could solve the equation this way–and after wade into fractions, but we prefer not have to do it. In systems in which substitution is not convenient, we will 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 a 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 and minus 1.

And again, in the next lesson we’ll talk about elimination.

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  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.

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