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Systems Of Linear Equations (Simultaneous Linear Equations)

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 .

Define System of Linear Equations

Note:

Every algebraic expression is written as:

is called a system of linear equations (In some references it is also called Simultaneous Linear Equations).

Whereas and are real numbers.

The solution of the system is the ordered pair .

Graphical Interpretation of a System of Linear Equations

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:

  1. 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.

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

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

Solving Systems of Linear Equations

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

Graphical Solution

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.

Example1:

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.

graphical solution of linear system of two equations

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:

Example2:

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.

graphical solution of linear system of two equations

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

Note:

We say that two lines are parallel if they have the same slope.

Example3:

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.

graphical solution of linear system of two equations

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

Note:

We say that two lines are coincide if they have the same slope and the same ordinate at the origin.

Algebraic Solution: Substitution Method

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

Note:

  1. Find an expression for one variable in terms of the other. (If one of the equations is already in this form, you can skip this step.)

  2. Substitute the expression found in step 1 into the other equation of the system. This will excludes one variable.

  3. Solve the equation containing one variable found in step 2.

  4. Back-substitute the value found in step 3 into one of the original equations of the system. Simplify and find the value of the remaining variable.

  5. Check the solution obtained in both of the system’s given equations.

Example4:

Solve the following system by the substitution method:

Solution:

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

Step1: Find an expression for one variable in terms of the other

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 .

Step2: Substitute the expression found in step 1 into the other equation of the system

Substitute in the equation (1):

The variable has been eliminated.

Step3: Solve the equation containing one variable found in step 2

We simplify and solve the equation (3):

Step4: Back-substitute

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

The solution of the system is the ordered pair .

Step5: Check the solution

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 .

Algebraic Solution: Elimination Method

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

Note:

  1. Choose the variable to be eliminated. (The coefficient is preferred 1 or -1 if exists)
  2. We multiply all the terms of one equation (or both in some cases) by a constant that will get the variable chosen in step 1 to add up to zero.
  3. Add the two equations together, in order to eliminate a variable, and then solve the simultaneous equation.
  4. Back-substitute the value found in step 3 into one of the original equations of the system. Simplify and find the value of the remaining variable.
  5. Check the solution obtained in both of the system’s given equations.

Example5:

Solve the following system by the elimination method:

Solution:

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

Step1: Choose the variable to be eliminated

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

Step2: Find the eequivalent system

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:

Step3: Adding the equations of the equivalent system

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

Step4: Back-substitute

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

The solution of the system is the ordered pair .

Step5: Check the solution

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 .