It is often desirable or even necessary to use more than one variable to model a situation in many fields. When this is the case, we write and solve a system of equations in order to answer questions about the situation.

If a system of linear equations has at least one solution, it is **
consistent**. If the system has no solutions, it is **inconsistent**. If
the system has an infinity number of solutions, it is **dependent**.
Otherwise it is **independent**.

A linear equation in three variables describes a plane and is an equation
equivalent to the equation

where A, B, C, and D are real numbers and A, B, C, and D are not all 0.

**Problem 3.1c:**

Your company has three acid solutions on hand: 30%, 40%, and 80% acid. It
can mix all three to come up with a 100-gallons of a 39% acid solution. If
it interchanges the amount of 30% solution with the amount of the 80%
solution in the first mix, it can create a 100-gallon solution that is 59%
acid. How much of the 30%, 40%, and 80% solutions did the company mix
to create a 100-gallons of a 39% acid solution?

**Answer:** 50 gallons of the 30% solution, 40 gallons of the
40% solution, and 10 gallons of the 80% solution.**
**

**
**

**
**

There are three unknowns:

Let's rewrite the paragraph that asks the question we are to answer.

The first sentence can be rewritten as [ The amount of 30% acid solution ] <*tex*2*html*_{c}*omment*_{m}*ark*>
+ [ The amount of 40% acid solution ] + [ The amount of 80% acid
solution ]
The second sentence can be rewritten 0.30 times [ The amount of money
invested in the safe investment ] + 0.40 times [ The amount of 40% acid
solution ] + 0.80 times [ The amount of 80% acid solution ]
The third sentence can be rewritten 0.80 times [ The amount of money
invested in the safe investment ] + 0.40 times [ The amount of 40% acid
solution ] + 0.30 times [ The amount of 80% acid solution ]

It is going to get boring if we keep repeating the phrases

Let's create a shortcut by letting symbols represent these phrases. Let

in the three sentences, and then rewrite them.

The first sentence [ The amount of 30% acid solution ] + [ The amount of
40% acid solution ] + [ The amount of 80% acid solution ] =100 can now
be written in the algebraic form

The second sentence 0.30 times [ The amount of money invested in the safe investment ] + 0.40 times [ The amount of 40% acid solution ] + 0.80 times [ The amount of 80% acid solution ] can now be written in the algebraic form

The third sentence 0.80 times [ The amount of money invested in the safe investment ] + 0.40 times [ The amount of 40% acid solution ] + 0.30 times [ The amount of 80% acid solution ] can now be written in the algebraic form

We have converted the problem from one described by words to one that is
described by three equations.

x+y+z |
= | 100 | (1) |

0.30x+0.40y+0.80z |
= | 39 | (2) |

0.80x+0.40y+0.30z |
= | 59 | (3) |

We are going to show you how to solve this system of equations three different ways:

1) Substitution,
2) Elimination
3) Matrices

**SUBSTITUTION**:

The process of substitution involves several steps:

Step 1: Solve for one of the variables in one of the equations. It
makes no difference which equation and which variable you choose. Let's
solve for *x* in equation (1).

Step 2: Substitute this value for

0.30x+0.40y+0.80z |
= | 39 | |

= | 39 | ||

30-0.30y-0.30z+0.40y+0.80z |
= | 39 | |

0.10y+0.50z |
= | 9 | |

y+5z |
= | 90 | (4) |

0.80x+0.40y+0.30z |
= | 59 | |

= | 59 | ||

80-0.80y-0.80z+0.40y+0.30z |
= | 59 | |

-0.40y-0.50z |
= | -21 | |

4y+5z |
= | 210 | (5) |

Step 3: Solve for

Step 4: Substitute this value of

Step 5: Substitute this value of

Step 7: Substitute 40 for

The solution is: and

Step 8: Check the solutions:

The process of elimination involves several steps: First you reduce three
equations to two equations with two variables, and then to one equation with
one variable.

Step 1: Decide which variable you will eliminate. It makes no
difference which one you choose. Let us eliminate *x* first.

Step 2: Multiply both sides of equation (1) by -0.30 and then add the transformed equation (1) to equation (2) to form equation (4).

Step 3: Multiply both sides of equation (1) by -0.80 and then add the transformed equation (1) to equation (3) to form equation (5).

Step : We now have two equations with two variables.

Step 4: Multiply both sides of equation (4) by -4 and add the transformed equation (4) to equation (5) to create equation (6) with just one variable.

Step 6: Substitute 10 for

Step 7: Substitute 10 for

The solution is: and

**MATRICES**:

The process of using matrices is essentially a shortcut of the process of
elimination. Each row of the matrix represents an equation and each column
represents coefficients of one of the variables.

Step 1:
Create a three-row by four-column matrix using coefficients and the constant
of each equation.

The vertical lines in the matrix stands for the equal signs between both
sides of each equation. The first column contains the coefficients of x, the
second column contains the coefficients of y, the third column contains the
coefficients of z, and the last column contains the constants.

We want to convert the original matrix

to the following matrix.

Because then you can read the matrix as

Step 2: We work with column 1 first. The number 1 is already in cell
11(Row1-Col 1). Add -0.30 times Row 1 to Row 2 to form a new Row 2, and
add -0.80 times Row 1 to Row 3 to form a new Row 3..

Step 3: We will now work with column 1. We want 1 in Cell 22, and we achieve this by multiplying Row 2 by 10 for a new Row 2.

Step 4: Let's now manipulate the matrix so that there are zeros in Cell 12 and Cell 32. We do this by adding -1 times Row 2 to Row 1 to form a new Row 1, and add 0.40 times Row 2 to Row 3 5o form a new Row 3.

Step 5: Let's now manipulate the matrix so that there is a 1 in Cell 33. We do this by multiplying Row 3 by

Step 6: Let's now manipulate the matrix so that there are zeros in Cell 13 and Cell 23. We do this by adding 4 times Row 3 to Row 1 for a new Row 1 and adding -5 times Row 3 to Row 3 for a new Row 3.

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 three by three system of equations,
click on
**Example**.

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**Author**: Nancy Marcus

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