# Methods of solving systems of linear equations

A **systems of equations** is a set of multiple equations that need to be solved. The goal is to find the unknown variable.

For example, we need a system of two equations to solve equations that have two variables. There arefour methods of solving systems of equations:

But first, lets remember:

It is valid:$\forall a, b, c$

- if $a<b$, then $\forall c\in \mathbb{R}$:

$a+c<b+c$ (1)

- if $a<b$, then $\forall c\in \mathbb{R}$:

$a \cdot c<b \cdot c$ (2)

- if $a<b$ and $b<c$

$a<c$ (3)

## Substitution method

Substitution is a simple method in which we solve one of the equations for one variable and then substitute that variable into the other equation and solve it.

$$x+y=4$$

$$ x-y=2$$

To solve the system of two equations above we need to follow these steps:

$$x+y=4 \Rightarrow y=4-x$$

$$ x-y=2$$

________________________

Now, we must solve this equation:

$x-(4-x)=2$

$ x-4+x=2$

we use property (1)

$$2x-4=2 /+4$$

$$2x-4+4=2+4$$

we use property (2)

$$2x=6 /:2$$

$x=3$

Now, we must return value of unknown variable of in first equation: $x+y=4$

$$3+y=4 /+(-3)$$

$3+y-3=4-3$

$y=1$

The result is: $(x,y)=(3,1)$

Now we can check our result.

$$x+y=4 \Rightarrow y=4-x,$$

$$x-y=2$$

________________

$$3+1=4$$

$$3-1=2$$

Let’s try to solve this system of equations using all methods.

## Graphing method

If we want to solve system of linear equation by this method, we must know that each linear equation is the line. That’s the reason that we call that function linear function. Now, we can easily draw that lines in coordinate plane.

For the line $x+y=5$:

For the line $x-y=1$:

We can see that the result is $(x,y)=(3,2)$. You can try to solve that problem by substitution method.

## Addition method

$$ x+y=5$$

$$ x-y=1$$

________

$$ x+y=5$$

$$ x-y=1$$

________

$$ 2x=6 /:3$$

$$ x=3$$

Now we return value of variable $x$ in one of the equations and we get:

$$ 3+y=5 /+(-3)$$

$$ 3+y-3=5-3$$

$$ y=2$$

The result is of system od equations is: $(x,y)=(3,2)$.

## Gaussian elimination method

We can solve once again system of linear equation:

$$ x+y=5$$

$$ x-y=1$$

___________

$$ x+y=5$$

$$ x-y=1 /\cdot (-1)$$

________

$$ x+y=5 $$

$$ -x+y=-1$$

$$ 2y=4 /:2$$

$$ y=2$$

Now,we can return variable $y$ in equation.

$$ x+2=5 /+(-2)$$

$x+2-2=5-2$

$x=3$

The result is of system of equations, once again, is: $(x,y)=(3,2)$.

To check if the results are right lets put our results for variable $x$ and variable $y$ in the first equation:

$$ 2 \cdot 2 + 2 \cdot 3 = 10$$

$ 4 + 6 = 10$

$ 10 = 10$

## Systems of equations worksheets

**Graphing - simple** (6.3 MiB, 1,122 hits)

**Graphing - advanced** (6.4 MiB, 1,066 hits)

**Substitution - simple** (148.3 KiB, 1,171 hits)

**Substitution - advanced** (143.0 KiB, 714 hits)

**Systems of equations - Elimination - Simple** (344.3 KiB, 489 hits)

**Systems of equations - Elimination - Advanced** (380.8 KiB, 359 hits)