#### 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:

2*x *+ 4*y *= 8

*x *+ 2*y *= 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 *+ 2*y *= -4

2*x *+ 4*y *= 8

**Solution:**

## Systems with Infinitely Many Solutions or No Solution

**Example 3:**

What is the solution to each system of equations?

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

**Solution:**

Substitute *y = *4* – *3*x* 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.

- {3x-y =-4}
- {y=3x-5 }

**Solution:**

Substitute *y = *3*x – *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

- When solving a system of equations using substitution, how can you determine whether the system has one solution, no solution, or infinitely many solutions?
- 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:

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