Given the linear system

ax + by = c

dx + ey = f

• A solution is an ordered pair

that will satisfy each equation

• The solution set is the set of all ordered pairs that satisfy

both equations.

## Another example:

Now, you try one:

Solve the system by graphing:

**2x+3 =y**

x+2y = -4

2x + 3 = y

x + 2y = -4

1. Solve one of the equations for either x or y.

2. Substitute that result into the other equation to obtain an equation in a

single variable (either x or y).

3. Solve the equation for that variable.

4. Substitute this value into any convenient equation to obtain the value of

the remaining variable.

## Another example:

**Solve the system using substitution:**

3x-2y=-7

y = 2x -3

• A consistent linear system is one that has one or more
solutions.

• A) If a consistent system has exactly one solution then the system is

said to be independent. E.g.

• What can you say about the slopes of the lines in this case?

• B) if a consistent system has more than one solution, then the

system is said to be dependent. E.g.

• What can you say about the slopes of the lines in this case?

• An inconsistent linear system is one that has no
solutions.

• E.g.

• What can you say about the slopes of the lines in this case?

2x – 7y=3

-5x + 3y = 7

A better method is elimination by addition: We may do any of the

following:

1) Two equations can be interchanged.

2. An equation is multiplied by a non -zero constant.

3. An equation is multiplied by a non- zero constant and then

added to another equation.

## Solve using elimination by addition

**Solve **

2x-5y = 6

-4x+10y=-1

## Applications

**A man walks at a rate of 3 miles per hour and jogs at a
rate of 5 miles**

per hour. He walks and jogs a total distance of 3.5 miles in 0.9

hours. How long does the man jog?

Let x= hours walked

y= hours jogged

## Now, solve the opening example:

A restaurant serves two types of fish dinners-small

for $5.99 each and a large order for $8.99.

One day, there were 134 total orders of fish and the total

receipts for these 134 orders was $1024.66. How many

small orders and how many large fish plates were

ordered?