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Chapter 5 Systems of Linear Equations

- Determine if the following ordered pair (point)

is a solution to the - following system of equations

Solution

How do you know if a point is a solution to these

two equations?

Plug the point into both equations to see if the

point makes both equations true. The following

will show this

Yes, 7 equals 7, so the point is a solution to

the 1st equation. Now lets try the 2nd equation.

Yes, 3 equals 3, so the point is also a solution

to the 2nd equation.

Since the point was a solution to BOTH equations,

the point is a solution to the system of

equations.

2. Determine if the following ordered pair

(point) is a solution to the following system

of equations

Solution

No, -10 does not equal -2, so the point is not a

solution to the 2nd equation.

Since the point was not a solution to BOTH

equations, the point is not a solution to the

system of equations.

Yes, -8 equals -8, so the point is a solution to

the 1st equation. Now lets try the 2nd equation.

3. Solve the following system of equations

graphically

Solution

We will graph each equation using the intercept

method from Chapter 4.

Again, the intercept method finds the

x-intercept (x,0), the y-intercept (0,y), and

a third point called the check point.

Lets graph the 1st equation

Lets plot these and draw the line

Again, well use the intercept method to graph

the 2nd equation

The solution to the system is (4,2)! This point

is common to both equations. That is, it is a

solution to both equations. Note This is the

point of intersection.

Again, lets plot these and draw the line (in

purple) above

4. Solve the following system of equations

graphically

Solution

Again, we will graph each equation using the

intercept method. Except this time, I will find

the x- and y-intercept but will not find the

check point.

Lets graph the 1st equation

And this line looks like

Again, well use the intercept method to graph

the 2nd equation

Uh-oh, there is no point that is common to both

lines these lines are parallel. The solution

is no solution. Note When the system has no

solution, the system is called inconsistent.

Again, lets plot these and draw the line (in

purple) above

5. Solve the following system of equations

graphically

Solution

Again, we will graph each equation using the

intercept method. Again, I will find the x- and

y-intercept but will not find the check point.

Lets graph the 1st equation

And this line looks like

Again, well use the intercept method to graph

the 2nd equation

Oh goodness, they are the same line. Thus, the

solution is all of the points on the line

(infinitely many ordered pair solutions). Note

When the system has all points on the line as a

solution, the system is called dependent.

Again, lets plot these and draw the line (in

purple) above

What are the advantages and disadvantages to

solving a system using the graphing method?

Advantage When you graph, you can SEE what is

happening with the linesare they intersecting?

are they parallel? are they the same line?

- Disadvantage
- If the solution is a fraction, it will be

difficult to write the answer from the graph.

For example, if the solution is , how

would you know this? - What happens if the point of intersection

happens at (1000,2000)- Is your graph that big? - What happens if the lines look parallel but

they really are not.

So, you can see some problems with the graphing

method

So heres the Substitution Method - an algebraic

method

6. Solve the following system of linear equations

by the Substitution Method

Solution

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7. Solve the following system of linear equations

by the Substitution Method

Solution

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8. Solve the following system of linear equations

by the Substitution Method

Solution

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- Solve the following system of linear equations by

the - Elimination Method

Solution

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10. Solve the following system of linear

equations by the Elimination Method

Solution

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11. Solve the following system of linear

equations by the Elimination Method

Solution

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Summary of Solving a System of Linear Equations

- You get a solution, (x,y). Graphically, this

means the lines intersect at this point (x,y).

This system is called consistent. - The variable terms cancel and you are left with a

false statement, such as 07. This means the

system has NO SOLUTION. Graphically, this means

the lines are parallel. Additionally, the system

is called inconsistent. - The variable terms cancel and you are left with a

true statement, such as 00. This means the

system has INFINITELY MANY SOLUTIONS (the points

on the line). Graphically, this means the lines

are the same. Additionally, the system is called

dependent.

Now for the Application Problems!

Solution

When the ferry travels downstream, the speed of

the ferry in still water is increased by the

speed of the current, hence the rate

When the ferry travels upstream, the speed of the

ferry in still water is decreased by the speed of

the current, hence the rate

Direction Rate Time Distance

Downstream 4

Upstream 6

Distance (from table) Distance (from problem)

Using the elimination method, multiply (a) by 3

and (b) by 2

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13. In the 2000 presidential election, George W.

Bush received 5 more electoral votes than Al

Gore. Together, the two presidential candidates

received a total of 537 electoral votes. How

many electoral votes did each receive? (Source

World Almanac and Book of Facts, 2002.)

Solution

Since George Bush received 5 more electoral votes

than Al Gore, one equation for the system of

linear equations is

Since the two candidates received a total of 537

electoral votes, the other equation for the

system of linear equations is

Thus, the system of linear equations to solve is

Since equation (a) is already solved for b, well

use substitution.

And since George W. Bush received 5 more

electoral votes, George W. Bush received

2665271 electoral votes.

Thus, Al Gore received 266 electoral votes.

14. In the Bay Head candy shop, double-dip mint

chocolates cost 10.00 per pound and swedish

fish cost 6.00 per pound. Mary, who loves

both, notices a mixture of the two selling for

7.00 per pound. She purchases 4 pounds and then

wonders how many pounds of each she received.

Solution

Since 4 pounds of the mixture were purchased, one

equation for the system of linear equations is

Now for the value (or cost) of each candy. If

the mints cost 10.00 per pound and x represents

the number of pounds you have, then the total

value (or cost) of the mints is

Using this same thinking, if the swedish fish

cost 6.00 per pound and y represents the number

of pounds you have, then the total value (or

cost) of the swedish fish is

Since the mixture was purchased at 7.00 per

pound and 4 pounds were purchased, the total cost

was (7.00)(4) 28.00 Thus the total value

equation leads to the other equation for the

system of linear equations

Thus, the system of linear equations to solve is

Substitution or elimination may be used to solve

this system. I chose elimination.

Note In the book, this problem was approached

using a table to set up the information.

Thus, 3 pounds of swedish fish were purchased.

And 1 pound of double-dip mint chocolates.

Okay, youre on your own You know what you need

to do!

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