Solving Systems of Linear Equations

We know that a linear equation is a function in which x and y have a linear relationship. If x increases, so too does y. We call this a linear equation because when we graph it on a Cartesian plane, it takes the form of a line with a given slope. But what exactly is a system of linear equations? More importantly, how do we solve these equations? Let's take a closer look at this concept and learn a few simple tricks to help us solve systems of linear equations:

What are systems of linear equations?

While a linear equation contains just one line, a system of linear equations contains many linear equations. A system of linear equations can contain 2 or more linear equations.

The possibilities of a system of linear equations

If we have two linear equations, then the graph of a system of two equations is a pair of lines on our Cartesian plane. In this case, there are three possibilities for us to consider:

• These two lines do not cross at all. There are no intersecting points, and the lines are parallel.
• The lines intersect at exactly one point -- and one point only. This is the most common outcome.
• The lines intersect at an infinite number of points. This means that the two lines occupy the same position. We can also say that they are the same line.

Systems of linear equations in which the lines do not cross

So what does it look like when lines do not intersect? As previously mentioned, this system takes the form of two parallel lines. Here's what it looks like:

In this equation, there are two linear equations:

$y=-2x+4$

$y=-2x-4$

As we can see, these two lines have an identical slope but a different horizontal displacement. This is what we can expect for all similar systems of linear equations with parallel lines. We also say that this system has "zero solutions" because there are no points at which the two lines intersect.

Systems of linear equations in which the lines cross once

As previously noted, the most common outcome with systems of linear equations is one in which the two lines cross once and only once. But what exactly does this look like?

Here we can see that there are two different linear equations:

$y=0.5x+2$

$y=-2x-3$

For our lines to be parallel, the two equations need to have very specific characteristics. But for lines to intersect, we can use almost all other values. The chances of the lines not being parallel are much higher. We say that this system of linear equations has "one solution" because the lines intersect once.

Systems of linear questions in which the lines are identical

The last possibility involves two identical lines that occupy exactly the same coordinates on the plane. Here's what that looks like:

If we look closely, we can see that the same coordinates are shared by both a red and a blue line. So how does this work? How can two linear equations be different and yet occupy the same coordinates? In this case, we're dealing with two equivalent equations:

$y=-2x-4$

$y+4=-2x$

Although these two equations are different, we can rearrange either one in such a way that it is identical to the second. Therefore, they are equivalent. We say that this type of system has "infinitely many solutions" because the two lines share an infinite number of points.

Methods for solving systems of linear equations

Solving systems of linear equations can be tricky, but there are a number of methods that can help us:

• The graphing method: This method is exactly what it sounds like. All we need to do is graph the two lines on a plane to find our solution. This method works best if we suspect that the two lines intersect at integer coordinates. It also works wonders if you're a visual learner.
• The substitution method: If you prefer to work things out using equations, the substitution method may be more suitable. We can start by solving one of the two linear equations for y in terms of x. Next, we can substitute that expression for y in the second linear equation. This will leave us with an equation in x. If we solve this equation, we will be left with the x-coordinate of the intersection. Finally, we can plug in our value for x in either equation to find the corresponding y-coordinate. Now we know exactly where the two lines intersect. Note that we can do this in reverse by instead solving the first equation for x in terms of y. We will get the same accurate result in the end.

Practicing solving systems of linear equations

Now that we have a few handy methods under our belts, we can practice our skills by solving some systems of linear equations.

Suppose we have these two linear equations:

• $3x+2y=16$
• $7x+y=19$

Together, they form a system of linear equations. So how would we "solve" this system? Remember that there are three possibilities:

• No solutions
• One solution
• An infinite number of solutions

With a cursory glance, we can see that these are not equivalent expressions. Therefore, an infinite number of solutions is unlikely. We can also see that the slopes are different -- which makes a zero-solution system equally unlikely. This means that the most likely outcome is one and only one intersection.

Let's use the substitution method:

We can start by solving the second equation for y:

$y=19-7x$

Now we can substitute this expression for y in the first equation:

$3x+2\left(19-7x\right)=16$

Let's solve for x:

$3x+38-14x=16$

$\left(-11x\right)=\left(-22\right)$

$x=2$

Great! Now we have the x-coordinate for the intersection point. But we still need the y-coordinate. Let's substitute x for 2 in the equation we created earlier when we solved for y:

$y=19-7x$

$y=19-7\left(2\right)$

$y=5$

Now that we know the y-coordinate, we are left with our coordinates for the single solution for this system of linear equations:

The pair of lines intersect once and only once at coordinates $\left(2,5\right)$ .

The linear combination method

But there's one more method we can use to solve systems of linear equations. This is called the linear combination method, although you might also hear names like the "addition method" or the "elimination method."

The concept is simple: We need to add a multiple of one equation to the other equation in such a way that we cancel the x terms or the y terms out. We can also do this by subtracting a multiple of one equation from the other in such a way that the x or y terms are canceled out. After we have accomplished this, all we need to do is solve for the remaining variable (either x or y). Finally, we can substitute to find the other coordinate value.

That might sound complicated, but it's actually quite straightforward when we see it in action. Let's try to solve a system of linear equations using the linear combination method:

• $4x+3y=-2$
• $8x-2y=12$

If we wanted to cancel out the x terms in the second equation, what would we need to multiply the first equation by? Let's try a multiple of (-2).

We are left with $-8x-6y=4$

Now we can add this to the second equation:

$-8x-6y=4$

Plus

$8x-2y=12$

Equals

$-8y=16$

Now that we have successfully canceled out the x term, it's time to solve for y:

$y=\left(-2\right)$

We can substitute this value for y in either of our first two equations to find the x-coordinate of our solution:

$4x+3\left(-2\right)=-2$

$4x-6=-2$

$4x=4$

$x=1$

Now we know that the solution for this system of linear equations is $\left(1,-2\right)$ .

Note that this method only works in certain situations. To find the most efficient method for solving systems of linear equations, we need to look closely at our two linear equations to spot opportunities for canceling out variable terms or substituting values.

Note that we can make the linear combination method even simpler by using a variation called the "matrix method."

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