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 2.1a:**

A total of $12,000 is invested in two funds paying 9% and 11% simple
interest. If the yearly interest is $1,180, how much of the $12,000 is
invested at 9% and how much is invested at 11%?

**Answer: $7,000 was invested at 9% and $5,000 was invested at
11%.
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Essentially, the first sentence is saying that ''...that the amount of money
invested at 9%...'' + ''...the amount of money invested at 11%...'' =
Essentailly, the second sentence is saying that ''...the amount of money
invested at 9%....''times''...the amount of money invested at
11%...''

It is going to get rather tiresome to keep repeating the two phrases ''...the amount of money invested at 9%....'' and ''...the amount of money invested at 11%...''. So let's rewrite the sentences in shorthand form. Let the symbol

The two sentences can now be written as

We have converted a narrative statement of the problem to an equivalent algebraic statement of the problem. 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 (1).

Step 2:

Substitute this value for *x* in equation (2). This will change equation (2)
to an equation with just one variable, *y*.

Step 3:

Solve for *y* in the translated equation (2).

Step 4:

Substitute this value of *y* in equation (1) and solve for *x*.

Step 5:

Check your answers by substituting the values of *x* and *y* in each of the
original equations. If, after the substitution, the left side of the
equation equals the right side of the equation, you know that your answers
are correct.

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 -.09 and add it to equation (2) to obtain
equation (3) with just one variable.

Step 2:

Substitute
in equation (1) and solve for *x*.

Step 3:

Check your answers.

**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. In
this case, we don't have to do anything. The number 1 is already in the
cell.

Step 2:

Add -0.09 times Row 1 to Row 2 to form a new Row 2.

Manipulate the matrix so that the cell 22 is 1. Do this by multiplying row 2
by 1/0.02.

**The method of Graphing**:

In this method solve for y in each equation and graph both. The point of
intersection is the 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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