Algebra:
Systems of Equations

1. 
\(\begin{array}{lcl}{5=x+y}\\{2y+x=7}\end{array}\)
 Solution
2. 
\(\begin{array}{lcl}{5x+{y}^{2}=7}\\{10x=8+{y}^{2}}\end{array}\)
 Solution
3. 
\(\begin{array}{lcl}{2x-3y=-2}\\{4x+y=24}\end{array}\)
 Solution
4. 
\(\begin{array}{lcl}{2x+y=8}\\{-6x-3y=10}\end{array}\)
 Solution
5. 
\(\begin{array}{lcl}{x+y-z=-2}\\{z+2x-y=5}\\{-x+2y+2z=1}\end{array}\)
 Solution

Systems Of Equations - Introduction

In algebra, a system of equations is a group of two or more equations that contain the same set of variables. A solution to the system is the values for the set of variables that can simultaneously satisfy all equations of the system.
When expressed graphically, since each equation of the system can be plotted as a line, when we are looking for the solution to the system, we are in fact looking for the intersection of these lines.
Also note that there are both “linear” and “non-linear” systems of equations. The difference is that linear equations yield straight lines and only contain only variables, coefficients and constants. Non-linear equations might contain exponents, square roots, etc.

Solving Systems Of Equations

It might seem obvious, but to meaningfully solve a system of equations, they must share one or more variables. For example, we can solve equations such as
\(5=x+y\)
and
\(2y+x=7\)
because they share the variables x and y.
How do we solve these equations? There are multiple methods, such as substitution, elimination, matrix, etc. For this tutorial, let us use substitution, which seems to be the most intuitive method to understand for beginners.
Solve the following system of equation:
\(5=x+y\)

\(2y+x=7\)
Let's start with the first equation,
\(5=x+y\)
. Subtracting y from both sides gives us
\(x\)
on its own as
\(5-y=x\)
, which can be rewritten as
\(x=5-y\)
.
Next, we substitute
\(x=5-y\)
into the second equation
\(2y+x=7\)
, which gives
\(2y+5-y=7\)
. We can simplify this to
\(y+5=7\)
, which tells us that
\(y=2\)
.
Then, we substitute
\(y=2\)
back into
\(x=5-y\)
, which is
\(x=5-2\)
. This simplifies to
\(x=3\)
.
We are done! We have found that
\(x=3\)
and
\(y=2\)
. This is the solution to this system of equations.

Next Steps

You can get more familiar with systems of equations using the practice problems at the top of this page. You can also try out other topics on our practice page. Ready to take your learning to the next level with “how” and “why” steps? Sign up for Cymath Plus today.