Suppose you buy a book for an amount where you give the seller $20 and they return $8. Can we know the price of the book? Of course, the matter is very simple, we can express this problem with a simple equation , where is the price of the book (One Unknown).

But now suppose you bought 3 books and 9 notebooks for $120. Can we know the price of one book and the price of one notebook? Of course not, this time it became a little more complicated. We can express this problem by the equation where is the price of the book and is the price of the book.

To solve this problem, we need another equation with the same unknowns as the first equation. For example .

the algebraic expression is an equation of a straight line that can be written in the form:. So, the system of two linear equations give us the equations of two straight lines (graphes) in a cartesian coordinate system. the relative positions of these two lines express the solution of the system:

The graphs intersect at exactly one point, which gives the (single) ordered-pair solution of the system. The system is consistent and the equations are independent.

The graphs are parallel lines, so there is no solution and the solution set is ∅. The system is inconsistent and the equations are independent.

The graphs are identical (lines coincide), and there are an infinitely many solutions. The system is consistent and the equations are dependent.

To solve a linear system of two equations, we have three different methods:

Write the equations of linear systems in the form of , and then represent the lines in the coordinate plane. As we explained earlier, these two lines can be intersecting, parallel, or coincident.

We will try to take an example for each of the three cases.

Let's solve the following system of linear equations:

We rewrite the two equations as follows in order to make it easier to represent them in the coordinate plane.

From the graph, we notice that the two lines intersect at the point with coordinates .

So the solution of the system is the ordered-pair .

The ordered-pair that means and .

**Check the solution**

We substitute and in the two equations:

Let's solve the following system of linear equations:

We rewrite the two equations as follows in order to make it easier to represent them in the coordinate plane.

From the graph, we notice that the two lines are parallel. Then there is no solution to the system.

Let's solve the following system of linear equations:

We rewrite the two equations as follows in order to make it easier to represent them in the coordinate plane.

From the graph, we notice that the two lines are identical. Then there is infinitely many solutions.

In a system of two linear equations with two variables, the **substitution method** involves converting the system to one equation in one variable by an appropriate substitution.

**Solving Linear Systems by Substitution**

Solve the following system by the substitution method:

Before solving, it is advisable to number the two equations:

We choose one of the unknowns in one of the two equations, write it in terms of the other unknown. If possible, solve for a variable whose coefficient is 1 or - 1 to avoid working with fractions as much as possible.

we choose the equation (2) and write in terms of because the coeffiecient of is .

Substitute in the equation (1):

The variable has been eliminated.

We simplify and solve the equation (3):

Now in Equation (1), we substitute into the value 4.

The solution of the system is the ordered pair .

Chek the solution in the both equations (1) and (2).

For equation (1):

For equation (2):

True statements result when the solution is substituted in both equations, confirming that the solution set is .

In a system of two linear equations with two variables, the **Elimination Method** involves finding the equivalent system which the coefficients of one of the two variables must be additive inverses.

**Solving Linear Systems by Elimination**

Solve the following system by the elimination method:

Before solving, it is advisable to number the two equations:

For example, we can choose to eliminat the variable because the coefficient is in the equation(1).

The coefficient of the variable in the equation (2) is 2, then we must multiply all terms of the equation (1) by **2** to get the additive inverses( and ).

The equivalent system is:

We add the equations (3) and (4) side by side:

Now in Equation (1), we substitute into the value -1.

The solution of the system is the ordered pair .

Chek the solution in the both equations (1) and (2).

For equation (1):

For equation (2):

True statements result when the solution is substituted in both equations, confirming that the solution set is .