Systems with Infinitely Many Solutions or No Solution

Introduction:

Systems with Infinitely Many Solutions:

A system of linear equations is consistent if it has one or more solutions.

Example 1:

Solve the following systems by graphing:

2x + 4y = 8

x + 2y = 4

Solution:

Systems with No Solution:

A system of linear equations is inconsistent if no solutions exist.

Example 2:

Solve the following systems by graphing:

x + 2y = -4

2x + 4y = 8

Solution:

Systems with Infinitely Many Solutions or No Solution

Example 3:

What is the solution to each system of equations?

1. { y = 4-3x}
2. { -6x-2y = -8 }

Solution:

Substitute y = 4 – 3x  into the second equation.

-6x – 2y =-8

-6x – 2(4-3x) = -8

-6x -8 + 6x = -8

-6x-8+ 6x =-8

-6x+ 6x -8 =-8

-8 = -8……. True Statement

The statement – 8 = – 8 is an identity, so the system of equations has infinitely many solutions. Both equations represent the same line. All points on the line are solutions to the system of equations.

1. {3x-y =-4}
2. {y=3x-5 }

Solution:

Substitute y = 3x – 5 into the first equation.

  3x-y= -4

3x-(3x-5) =-4

3x-3x+5= -4

5=-4…….. False Statement

The statement 5 = –4  is false, so the system of equations has no solution.

Example 4:

What is the solution to each system of equations?

x + y = –4

y = –x + 5

Solution:

Substitute y = –x + 5 into the first equation.

x + y = –4

x –x + 5 = –4

5 = –4…………False statement.

The statement 5 = –4  is false, so the system of equations has no solution.

y = –2x + 5

2x + y = 5

Solution:

Substitute y = –2x + 5 into the second equation.

2x + y = 5

2x + (–2x + 5) = 5

2x –2x + 5 = 5

5 = 5…………….True statement.

The statement 5 = 5 is an identity, so the system of equations has infinitely many solutions. Both equations represent the same line. All points on the line are solutions to the system of equations.

Model Using Systems of Equations

Example 5:

Funtime Amusement Park charges $12.50 for admission and then $0.75 per ride. River’s Edge Park charges $18.50 for admission and then $0.50 per ride. For what number of rides is the cost the same at both parks?

Solution:

Formulate:

Write a system of linear equations to model the cost of both parks.

In both equations, let y represent the dollar amount of charges. Let x represent the number of rides.

y = 0.75x + 12.5

y = 0.5x + 18.5

Compute:

Substitute for y in one of the equations.

y = 0.75x + 12.5

0.5x + 18.5 = 0.75x + 12.5

0.5x + 18.5 – 12.5 = 0.75x

0.5x + 6 = 0.75x

6 = 0.75x – 0.5x

6 = 0.25x

x = 6/0.25

x = 24

Interpret:

Since x is the number of rides, for 24 rides the cost will be the same at both parks.

Example 6:

At a hot air balloon festival, Mohamed’s balloon is at an altitude of 40 m and rises at 10 m/min. Dana’s balloon is at an altitude of 165 m and descends at 15 m/min. In how many minutes will both balloons be at the same altitude?

Solution:

Formulate:

Write a system of linear equations to model the cost of both parks.

In both equations, let y represent the altitude of the balloon. Let x represent the number of minutes.

y = 40 + 10x

y = 165 – 15x

Compute:

Substitute for y in one of the equations.

y = 40 + 10x

165 – 15x = 40 + 10x

165 = 40 + 10x + 15x

165 = 40+ 25x

x = 125/25

x = 5

Interpret:

Since x is the number of minutes, it takes 5 minutes for both balloons to be at the same altitude.

Exercise

1. When solving a system of equations using substitution, how can you determine whether the system has one solution, no solution, or infinitely many solutions?
2. Use substitution to solve the following system of equations.

4x + 8y = -8

x = -2y + 1

3. Use substitution to solve the following system of equations.

2x – 3y = 6

y = 2/3 x  – 2

4. When given a system of equations in slope-intercept form, which is the most efficient method to solve: graphing or substitution? Explain.

5. Use substitution to solve the following system of equations.

2x + 2y = 6

4x + 4y = 4

6. Use substitution to solve the following system of equations.

2x + 5y = -5

y = – 2/5 x – 1

7. Richard and Teo have a combined age of 31. Richard is 4 years older than twice Teo’s age. How old are Richard and Teo?

8. Abby uses two social media sites. She has 52 more followers on site A than on site B. How many followers does she have on each site?

9. Use substitution to solve the following system of equations.

4x + 2y = −3

2x + y = 1

10. Stay Fit gym charges a membership fee of $75. They offer karate classes for an additional fee.

How many classes could members and non-members take before they pay the same amount?

Concept Map: