Systems of Linear Equations (2024)

Systems of Linear Equations (1)
A Linear Equation is an equation for a line.

A linear equation is not always in the form y = 3.5 − 0.5x,

It can also be like y = 0.5(7 − x)

Or like y + 0.5x = 3.5

Or like y + 0.5x − 3.5 = 0 and more.

(Note: those are all the same linear equation!)

A System of Linear Equations is when we have two or more linear equations working together.

Example: Here are two linear equations:

2x+y=5
−x+y=2

Together they are a system of linear equations.

Can you discover the values of x and y yourself? (Just have a go, play with them a bit.)

Let's try to build and solve a real world example:

Example: You versus Horse

Systems of Linear Equations (2)

It's a race!

You can run 0.2 km every minute.

The Horse can run 0.5 km every minute. But it takes 6minutes to saddle the horse.

How far can you get before the horse catches you?

We can make two equations (d=distance in km, t=time in minutes)

  • You run at 0.2km every minute, so d = 0.2t
  • The horse runs at 0.5 km per minute, but we take 6 off its time: d = 0.5(t−6)

So we have a system of equations (that are linear):

  • d = 0.2t
  • d = 0.5(t−6)

We can solve it on a graph:

Systems of Linear Equations (3)

Do you see how the horse starts at 6 minutes, but then runs faster?

It seems you get caught after 10 minutes ... you only got 2 km away.

Run faster next time.

So now you know what a System of Linear Equations is.

Let us continue to find out more about them ....

Solving

There can be many ways to solve linear equations!

Let us see another example:

Example: Solve these two equations:

Systems of Linear Equations (4)

  • x + y = 6
  • −3x + y = 2

The two equations are shown on this graph:

Our task is to find where the two lines cross.

Well, we can see where they cross, so it is already solved graphically.

But now let's solve it using Algebra!

Hmmm ... how to solve this? There can be many ways! In this case both equations have "y" so let's try subtracting the whole second equation from the first:

x + y − (−3x + y) = 6 − 2

Now let us simplify it:

x + y + 3x − y = 6 − 2

4x = 4

x = 1

So now we know the lines cross at x=1.

And we can find the matching value of y using either of the two original equations (because we know they have the same value at x=1). Let's use the first one (you can try the second one yourself):

x + y = 6

1 + y = 6

y = 5

And the solution is:

x = 1 and y = 5

And the graph shows us we are right!

Linear Equations

Only simple variables are allowed in linear equations. No x2, y3, √x, etc:

Systems of Linear Equations (5)
Linear vs non-linear

Dimensions

A Linear Equation can be in 2 dimensions ...
(such as x and y)
Systems of Linear Equations (6)
... or in 3 dimensions ...
(it makes a plane)
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... or 4 dimensions ...
... or more!

Common Variables

Equations that "work together" share one or more variables:

A System of Equations has two or more equations in one or more variables

Many Variables

So a System of Equations could have many equations and many variables.

Example: 3 equations in 3 variables

2x+y2z=3
xyz=0
x+y+3z=12

There can be any combination:

  • 2 equations in 3 variables,
  • 6 equations in 4 variables,
  • 9,000 equations in 567 variables,
  • etc.

Solutions

When the number of equations is the same as the number of variables there is likely to be a solution. Not guaranteed, but likely.

In fact there are only three possible cases:

  • No solution
  • One solution
  • Infinitely many solutions

When there is no solution the equations are called "inconsistent".

One or infinitely many solutions are called "consistent"

Here is a diagram for 2 equations in 2 variables:

Systems of Linear Equations (8)

Independent

"Independent" means that each equation gives new information.
Otherwise they are "Dependent".

Also called "Linear Independence" and "Linear Dependence"

Example:

  • x + y = 3
  • 2x + 2y = 6

Those equations are "Dependent", because they are really the same equation, just multiplied by 2.

So the second equation gave no new information.

Where the Equations are True

The trick is to find where all equations are true at the same time.

True? What does that mean?

Example: You versus Horse

Systems of Linear Equations (9)

The "you" line is true all along its length (but nowhere else).

Anywhere on that line d is equal to 0.2t

  • at t=5 and d=1, the equation is true (Is d = 0.2t? Yes, as 1 = 0.2×5 is true)
  • at t=5 and d=3, the equation is not true (Is d = 0.2t? No, as 3 = 0.2×5 is not true)

Likewise the "horse" line is also true all along its length (but nowhere else).

But only at the point where they cross (at t=10, d=2) are they both true.

So they have to be true simultaneously ...

... that is why some people call them "Simultaneous Linear Equations"

Solve Using Algebra

It is common to use Algebra to solve them.

Here is the "Horse" example solved using Algebra:

Example: You versus Horse

The system of equations is:

  • d = 0.2t
  • d = 0.5(t−6)

In this case it seems easiest to set them equal to each other:

d = 0.2t = 0.5(t−6)

Start with:0.2t = 0.5(t − 6)

Expand 0.5(t−6):0.2t = 0.5t − 3

Subtract 0.5t from both sides:−0.3t = −3

Divide both sides by −0.3:t = −3/−0.3 = 10 minutes

Now we know when you get caught!

Knowing t we can calculate d:d = 0.2t = 0.2×10 = 2 km

And our solution is:

t = 10 minutes and d = 2 km

Algebra vs Graphs

Why use Algebra when graphs are so easy? Because:

More than 2 variables can't be solved by a simple graph.

So Algebra comes to the rescue with two popular methods:

  • Solving By Substitution
  • Solving By Elimination

We will see each one, with examples in 2 variables, and in 3 variables. Here goes ...

Solving By Substitution

These are the steps:

  • Write one of the equations so it is in the style "variable = ..."
  • Replace (i.e. substitute) that variable in the other equation(s).
  • Solve the other equation(s)
  • (Repeat as necessary)

Here is an example with 2 equations in 2 variables:

Example:

  • 3x + 2y = 19
  • x + y = 8

We can start with any equation and any variable.

Let's use the second equation and the variable "y" (it looks the simplest equation).

Write one of the equations so it is in the style "variable = ...":

We can subtract x from both sides of x + y = 8 to get y = 8 − x. Now our equations look like this:

  • 3x + 2y = 19
  • y = 8 − x

Now replace "y" with "8 − x" in the other equation:

  • 3x + 2(8 − x) = 19
  • y = 8 − x

Solve using the usual algebra methods:

Expand 2(8−x):

  • 3x + 16 − 2x = 19
  • y = 8 − x

Then 3x−2x = x:

  • x + 16 = 19
  • y = 8 − x

And lastly 19−16=3

  • x = 3
  • y = 8 − x

Now we know what x is, we can put it in the y = 8 − x equation:

  • x = 3
  • y = 8 − 3 = 5

And the answer is:

x = 3
y = 5

Note: because there is a solution the equations are "consistent"

Check: why don't you check to see if x = 3 and y = 5 works in both equations?

Solving By Substitution: 3 equations in 3 variables

OK! Let's move to a longer example: 3 equations in 3 variables.

This is not hard to do... it just takes a long time!

Example:

  • x + z = 6
  • z − 3y = 7
  • 2x + y + 3z = 15

We should line up the variables neatly, or we may lose track of what we are doing:

x+z=6
3y+z=7
2x+y+3z=15

We can start with any equation and any variable. Let's use the first equation and the variable "x".

Write one of the equations so it is in the style "variable = ...":

x=6 − z
3y+z=7
2x+y+3z=15

Now replace "x" with "6 − z" in the other equations:

(Luckily there is only one other equation with x in it)

x=6 − z
3y+z=7
2(6−z)+y+3z=15

Solve using the usual algebra methods:

2(6−z) + y + 3z = 15 simplifies to y + z = 3:

x=6 − z
3y+z=7
y+z=3

Good. We have made some progress, but not there yet.

Now repeat the process, but just for the last 2 equations.

Write one of the equations so it is in the style "variable = ...":

Let's choose the last equation and the variable z:

x=6 − z
3y+z=7
z=3 − y

Now replace "z" with "3 − y" in the other equation:

x=6 − z
3y+3 − y=7
z=3 − y

Solve using the usual algebra methods:

−3y + (3−y) = 7 simplifies to −4y = 4, or in other words y = −1

x=6 − z
y=−1
z=3 − y

Almost Done!

Knowing that y = −1 we can calculate that z = 3−y = 4:

x=6 − z
y=−1
z=4

And knowing that z = 4 we can calculate that x = 6−z = 2:

x=2
y=−1
z=4

And the answer is:

x = 2
y = −1
z = 4

Check: please check this yourself.

We can use this method for 4 or more equations and variables... just do the same steps again and again until it is solved.

Conclusion: Substitution works nicely, but does take a long time to do.

Solving By Elimination

Elimination can be faster ... but needs to be kept neat.

"Eliminate" means to remove: this method works by removing variables until there is just one left.

The idea is that we can safely:

  • multiply an equation by a constant (except zero),
  • add (or subtract) an equation on to another equation

Like in these two examples:

Systems of Linear Equations (10)

CAN we safely add equations to each other?

Yes, because we are "keeping the balance".

Imagine two really simple equations:

x − 5 = 3
5 = 5

We can add the "5 = 5" to "x − 5 = 3":

x − 5 + 5 = 3 + 5
x = 8

Try that yourself but use 5 = 3+2 as the 2nd equation

It works just fine, because both sides are equal (that is what the = is for)

We can also swap equations around, so the 1st could become the 2nd, etc, if that helps.

OK, time for a full example. Let's use the 2 equations in 2 variables example from before:

Example:

  • 3x + 2y = 19
  • x + y = 8

Very important to keep things neat:

3x+2y=19
x+y=8

Now ... our aim is to eliminate a variable from an equation.

First we see there is a "2y" and a "y", so let's work on that.

Multiply the second equation by 2:

3x+2y=19
2x+2y=16

Subtract the second equation from the first equation:

x=3
2x+2y=16

Yay! Now we know what x is!

Next we see the 2nd equation has "2x", so let's halve it, and then subtract "x":

Multiply the second equation by ½ (i.e. divide by 2):

x=3
x+y=8

Subtract the first equation from the second equation:

x=3
y=5

Done!

And the answer is:

x = 3 and y = 5

And here is the graph:

Systems of Linear Equations (11)

The blue line is where 3x + 2y = 19 is true

The red line is where x + y = 8 is true

At x=3, y=5 (where the lines cross) they are both true. That is the answer.

Here is another example:

Example:

  • 2x − y = 4
  • 6x − 3y = 3

Lay it out neatly:

2xy=4
6x3y=3

Multiply the first equation by 3:

6x3y=12
6x3y=3

Subtract the second equation from the first equation:

00=9
6x3y=3

0 − 0 = 9 ???

What is going on here?

Quite simply, there is no solution.

They are actually parallel lines:Systems of Linear Equations (12)

And lastly:

Example:

  • 2x − y = 4
  • 6x − 3y = 12

Neatly:

2xy=4
6x3y=12

Multiply the first equation by 3:

6x3y=12
6x3y=12

Subtract the second equation from the first equation:

00=0
6x3y=3

0 − 0 = 0

Well, that is actually TRUE! Zero does equal zero ...

... that is because they are really the same equation ...

... so there are an Infinite Number of Solutions

They are the same line:Systems of Linear Equations (13)

And so now we have seen an example of each of the three possible cases:

  • No solution
  • One solution
  • Infinitely many solutions

Solving By Elimination: 3 equations in 3 variables

Before we start on the next example, let's look at an improved way to do things.

Follow this method and we are less likely to make a mistake.

First of all, eliminate the variables in order:

  • Eliminate xs first (from equation 2 and 3, in order)
  • then eliminate y (from equation 3)

Start with:

Systems of Linear Equations (14)

Eliminate in this order:

Systems of Linear Equations (15)

We then have this "triangle shape":

Systems of Linear Equations (16)

Now start at the bottom and work back up (called "Back-Substitution")
(put in z to find y, then z and y to find x):

Systems of Linear Equations (17)

And we are solved:

Systems of Linear Equations (18)

ALSO, it is easier to do some of the calculations in our head, or on scratch paper, instead of always working within the set of equations:

Example:

  • x + y + z = 6
  • 2y + 5z = −4
  • 2x + 5y − z = 27

Written neatly:

x+y+z=6
2y+5z=−4
2x+5yz=27

First, eliminate x from 2nd and 3rd equation.

There is no x in the 2nd equation ... move on to the 3rd equation:

Subtract 2 times the 1st equation from the 3rd equation (just do this in your head or on scratch paper):

Systems of Linear Equations (19)

And we get:

x+y+z=6
2y+5z=−4
3y3z=15

Next, eliminate y from 3rd equation.

We could subtract 1½ times the 2nd equation from the 3rd equation (because 1½ times 2 is 3) ...

... but we can avoid fractions if we:

  • multiply the 3rd equation by 2 and
  • multiply the 2nd equation by 3

and then do the subtraction ... like this:

Systems of Linear Equations (20)

And we end up with:

x+y+z=6
2y+5z=−4
z=−2

We now have that "triangle shape"!

Now go back up again "back-substituting":

We know z, so 2y+5z=−4 becomes 2y−10=−4, then 2y=6, so y=3:

x+y+z=6
y=3
z=−2

Then x+y+z=6 becomes x+3−2=6, so x=6−3+2=5

x=5
y=3
z=−2

And the answer is:

x = 5
y = 3
z = −2

Please check this for yourself, it is good practice.

General Advice

Once you get used to the Elimination Method it becomes easier than Substitution, because you just follow the steps and the answers appear.

But sometimes Substitution can give a quicker result.

  • Substitution is often easier for small cases (like 2 equations, or sometimes 3 equations)
  • Elimination is easier for larger cases

And it always pays to look over the equations first, to see if there is an easy shortcut ... so experience helps ... so get experience!

Pencils and Jars Puzzle

591, 592, 593, 594, 1240, 61, 1241, 2863, 8157, 8158

Linear Equations Algebra Index

Systems of Linear Equations (2024)

FAQs

What are the possible answers for a system of linear equations? ›

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases. One solution. A system of linear equations has one solution when the graphs intersect at a point.

How to find out how many solutions a system of equations has? ›

A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

What's the easiest way to solve systems of linear equations? ›

SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING.
  1. Graph the first equation.
  2. Graph the second equation on the same rectangular coordinate system.
  3. Determine whether the lines intersect, are parallel, or are the same line.
  4. Identify the solution to the system. ...
  5. Check the solution in both equations.
Nov 24, 2022

How many answers can a linear equation have? ›

A linear equation could have exactly 1, 0, or infinite solutions. If we can solve the equation and get something like x=b where b is a specific number, then we have one solution.

What are the three possible solutions to a system of equations? ›

The three possible solutions to a system of equations are one solution, infinite solutions, or no solutions. One solution means a single point satisfies the system. Infinite solutions mean an infinite number of points satisfy the system. No solution means that no points satisfy the system.

What is not a possible solution to a system of linear equations? ›

If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that is true for both equations.

How to know if a system has no solutions? ›

A system of two linear equations has no solution if the lines are parallel. Parallel lines on a coordinate plane have the same slope and different y-intercepts (see figure 3 for an example of this).

How to know if a system of equations has infinite solutions? ›

The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept.

How to tell if a system of equations has one solution without graphing? ›

If the slope of the lines is different, then the system of equations has one solution. Therefore, you can determine that a system of linear equations has one solution without graphing by comparing the slope of the lines.

What is the fastest way to solve linear equations? ›

Substitution Method of Solving Linear Equations

To solve a linear equation using the substitution method, first, isolate the value of one variable from any of the equations. Then, substitute the value of the isolated variable in the second equation and solve it. Take the same equations again for example.

What are the three methods for solving systems of equations? ›

There are three ways to solve a system of linear equations: graphing, substitution, and elimination. The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system. The solution is the ordered pair(s) common to all lines in the system when the lines are graphed.

How to find solutions of linear equations? ›

The steps for solving linear equations are:
  1. Simplify both sides of the equation and combine all same-side like terms.
  2. Combine opposite-side like terms to obtain the variable term on one side of the equal sign and the constant term on the other.
  3. Divide or multiply as needed to isolate the variable.
  4. Check the answer.
Oct 6, 2021

Can you always check your answers when solving a linear equation? ›

With any equation, you can check your solution by substituting the value for the variable in the original equation. In other words, you evaluate the original equation using your solution. If you get a true statement, then your solution is correct.

Can linear equations have 2 answers? ›

Most linear systems you will encounter will have exactly one solution. However, it is possible that there are no solutions, or infinitely many. (It is not possible that there are exactly two solutions.) The word unique in this context means there is a solution, and it's the only one.

What is the infinite number of solutions? ›

An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If you simplify the equation using an infinite solutions formula or method, you'll get both sides equal, hence, it is an infinite solution. Infinite represents limitless or unboundedness.

What are the possibilities of a solution of a system of linear equations? ›

A linear system may behave in any one of three possible ways: The system has infinitely many solutions. The system has a unique solution. The system has no solution.

How to identify solutions to a system of linear equations? ›

To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions.

What are the types of solutions for the system of linear equations? ›

Solution of a System of Linear Equation

Linear equations can have three kinds of possible solutions: No Solution. Unique Solution. Infinite Solution.

How many solutions can a system of inequalities have? ›

A linear system of inequalities has an infinite number of solutions. Recall that when graphing a linear inequality the solution is a shaded region of the graph which contains all the possible solutions to the inequality.

References

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