Math Rocks

Chapter 7.1:

Two equations, such as y = 69 - 6.9x and y = 5.7 + 6.3x, together are called a system of equations. A solution of a system of equations is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have 0, 1, or an infinite number of solutions.

If the graphs intersect or coincide, the system of equations is said to be consistent. That is, it has at least one ordered pair that satisfies both equations.
If the graphs are parallel, the system of equations is said to be inconsistent. There are no ordered pairs that satisfy both equations.
Consistent equations can be independent or dependent. If a system has exactly one solution, it is independent. If the system has an infinite number of solutions, it is dependent.

1) y = 36 – 9x

3x + ^y/₃ = 12

Process: 3x + ^y/₃ = 12

9x + y = 36

y = 36 – 9x

2) 7x + 2y = 16

–21x – 6y = 24

Process:

7x + 2y = 16

2y = –7x + 16

y = –( ⁷/₂)x + 8

–21x – 6y = 24

–21x – 24 = 6y

–( ²¹/₆ )x – 4 = y

–( ⁷/₂)x – 4 = y

Chapter 7.2

Substitution

The exact solution of a system of equations can be found by suing algebraic methods. One such method is called substitution.

Steps for Using the Substitution Method in order to Solve Systems of Equations

Solve 1 equation for 1 variable. (Put in y = or x = form)

Substitute this expression into the other equation and solve for the missing variable.

Substitute your answer into the first equation and solve.

Check the solution.

These directions will make a lot more sense when you study the examples below!

Example 1

substitution method

The next example demonstrates a situation where it is easier to solve for x initially.

Example 2

substitution method for solving a system of equations

Let's take a look at another example. You'll find this very interesting!

Example 3

solving systems of equations

Ok... one more unique example!

Example 4

substitution method example