### Case 2

The lines do not intersect. When the lines do not intersect, you have no solutions (Ø).

### Case 3

The lines coincide. When the lines coincide, you have an infinite number of solutions.

### Graphing a System of Linear Equations

To graph a system of linear equations, use what you already know about graphing linear equations. To graph a system of linear equations, you will use the slope-intercept form of graphing. The first step is to transform the equations into slope-intercept form, or *y = mx + b*. Then use the slope and *y*-intercept to graph the line. Once you have both lines graphed, determine your solutions.

**Example:**

*x – y* = 6

2*x + y* = 3

Transform the first equation into y = mx + b. |
x – y = 6 |

Subtract x from both sides of the equation. |
x – x – y = 6 – x |

Simplify. | –y = 6 – x |

Use the commutative property. | –y = –x + 6 |

Multiply both sides of the equation by –1. | –1 · –y = –1(–x + 6) |

Simplify both sides. | y = x – 6 |

Transform the second equation into y = mx + b. |
2x + y = 3 |

Subtract 2x from both sides of the equation. | 2x – 2x + y = 3 – 2x |

Simplify. | y = 3 – 2x |

Use the commutative property. | y = –2x + 3 |

The slope of the first equation is 1, and the *y*-intercept is –6. The slope of the second equation is –2, and the *y*-intercept is 3. In the first equation, the line cuts the *y*-axis at –6. From that point, go up 1 and right 1. Draw a line through your beginning point and the endpoint—this line can extend as long as you want in both directions since it is endless. In the second equation, the line cuts the *y*-axis at 3. From that point, go down 2 and to the right 1. Draw a line through your beginning point and the endpoint, extending it as long as you want. The point of intersection of the two lines is (3,–3), so there is one solution.

**Example:**

*x* + 2*y* = –4

2*x* + 4*y* = –8

Transform the first equation into y = mx + b. |
x + 2y = –4 |

Subtract x from both sides of the equation. | x – x + 2y = –4 – x |

Simplify. | 2y = –4 – x |

Use the commutative property. | 2y = –x – 4 |

Divide both sides of the equation by 2. | |

Simplify both sides of the equation. | |

Transform the second equation into y = mx + b. |
2x + 4y = –8 |

Subtract 2x from both sides of the equation. | 2x – 2x + 4y = –8 – 2x |

Simplify. | 4y = –8 – 2x |

Use the commutative property. | 4y = –2x – 8 |

Divide both sides of the equation by 4. | |

Simplify both sides of the equation. |

The first and second equations transformed into the same equation, . If you have the same equation, it will graph into the same line. This is a case where the two lines coincide. Graph the line. Start with the *y*-intercept, which is –2. From that point, go down 1 and to the right 2. Since the lines coincide, there are an infinite number of solutions.

Here's a time saver! You can determine the nature of the solutions of a system without actually graphing them. Once you have the equations transformed into *y = mx + b*, compare the slopes and *y*-intercepts of each equation. If the slopes and *y*-intercepts are the same, you will have the same line. If the slopes are the same, but the *y*-intercepts are different, then the lines will be parallel. If the slopes are different, then the lines will intersect.

**Case 1:** The slopes and *y*-intercepts are the same, so the lines coincide. There will be an infinite number of solutions.

y= 2x+ 3y= 2x+ 3

**Case 2:** The slopes are the same, but the y-intercepts are different. The lines will be parallel, so there is no solution.

y= 2x+ 4y= 2x– 5

**Case 3:** The slopes are different. The lines will intersect, so there will be one solution.

y= 3x+ 2y= 2x+ 3

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