A system of equations is a collection of two or more equations with the same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.

The equations in the system can be linear or non-linear. This tutorial
reviews systems of linear equations.

A problem can be expressed in narrative form or the problem can be expressed
in algebraic form.

Let's start with an example stated in narrative form. We'll convert it to an
equivalent equation in algebraic form, and then we will solve it.

**Problem 2.1e:**

Solve for x and y in the following system of equations.

**Answer: No solution.
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Let's solve this system of equations.

A system of linear equations can be solved four different ways:**
Substitution
Elimination
Matrices
Graphing
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**The Method of Substitution:**

The method of substitution involves several steps:

Step 1:

Solve for *x* in equation (2).

Step 2:

Substitute this value for *x* in equation (1). This will change equation (1)
to an equation with just one variable, *y*.

Step 3:

Solve for *y* in the translated equation (1).

Step 4:

There is no solution.

**The Method of Elimination:**

The process of substitution involves several steps:

In a two-variable problem rewrite the equations so that when the equations
are added, one of the variables is eliminated, and then solve for the
remaining variable.

Step 1:

Multiply equation (1) by -5 and add it to equation (2) to form equation
(3) with just one variable.

Step 2:

When we added the changed, but equivalent equation (1) to equation (2), the
result was a false statement. What does this mean? It means there is no
unique solution to this problem.

**The Method of Matrices**:

This method is essentially a shortcut for the method of elimination.

Rewrite equations (1) and (2) without the variables and operators. The left
column contains the coefficients of the *x*'s, the middle column contains
the coefficients of the *y*'s, and the right column contains the
constants.

The objective is to reorganize the original matrix into one that looks like

Step 1.

Manipulate the matrix so that the number in cell 11 (row 1-col 1) is 1. To
achieve this, multiply Row 1 by

Manipulate the matrix so that the number is cell 21 (row 2-col 1) is 0. Do
this by adding -9 times Row 1 to Row 2 to form a new Row 2.

The last row of the matrix states that 0=-17, which is a false statement.
The conclusion is that there is no solution to this system.

**The method of Graphing**:

Solve for y in each equation and graph. Note that the two lines never
intersect. Therefore, there is no solution.

If you would like to go back to the problem page, click on
**Problem**.

If you would like to review the solution to the next problem, click on
**Problem**

If you would like to return to the beginning of the two by two system of equations,
click on
**Example**.

**
This site was built to accommodate the needs of students. The topics and problems are what
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**Author**: Nancy Marcus

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