When we have an equation with two **different** unknowns, like y = 2x + 1, we cannot solve the equation.

We can instead find pairs of x and y values that make the left side equal the right side

We can use a table of values to show the number pairs:

x |
0 | 1 | 2 | 3 |

y |

To complete this table of values, for y = 2x + 1, we need to find out the value of y when x = 0, when x = 1, when x = 2 and when x = 3

We do this using substitution

When x = 0

y = 2(0) + 1

y = 1

When x = 1

y = 2(1) + 1

y = 3

When x = 2

y = 2(2) + 1

y = 5

When x = 3

y = 2(3) + 1

y = 7

We can use these values to complete the table

x |
0 | 1 | 2 | 3 |

y |
1 | 3 | 5 | 7 |

**Example 1: Complete the table of values for x + y = 6**

x |
0 | 1 | 2 | 3 |

y |

We can find the y values using substitution:

When x = 0

0 + y = 6

y = 6

When x = 1

0 + y = 6

y = 5

When x = 2

0 + y = 6

y = 4

When x = 3

0 + y = 6

y = 3

We can now complete the table:

x |
0 | 1 | 2 | 3 |

y |
6 | 5 | 4 | 3 |

**Example 2: Complete the table of values for y = 3x - 5**

x |
-3 | -2 | -1 | 0 | 1 | 2 | 3 |

y |

In this example we have negative x values in the table. It is often easier to start with the positive x values:

When x = 3

y = 3(3) - 5

y = 4

When x = 2

y = 3(2) - 5

y = 1

When x = 1

y = 3(1) - 5

y = -2

When x = 0

y = 3(0) - 5

y = -5

x |
-3 | -2 | -1 | 0 | 1 | 2 | 3 |

y |
-5 | -2 | 1 | 4 |

We can notice a pattern in these y values, they are going up by 3 each time (from left to right or down by 3 each time from right to left).

We can expect that for x = -1, y = -8

When x = -2, y = -11

and when x = -3, y = -14

Substituting gives:

When x = -1

y = 3(-1) - 5

y = -8

When x = -2

y = 3(-2) - 5

y = -11

When x = -3

y = 3(-3) - 5

y = -14

If we make a mistake in our calculations we will be able to notice as our values would not fit the pattern.

x |
-3 | -2 | -1 | 0 | 1 | 2 | 3 |

y |
-14 | -11 | -8 | -5 | -2 | 1 | 4 |

**Try these:**

We can use a table of values to draw a graph. Each pair of values become a set of coordinates (x,y).

**Example: Here is the table of values for y = 2x - 3draw the graph for y = 2x - 3**

x |
-3 | -2 | -1 | 0 | 1 | 2 | 3 |

y |
-9 | -7 | -5 | -3 | -1 | 1 | 3 |

On a graph we will plot the coordinates:

(-3,-9), (-2,-7), (-1,-5), (0,-3), (1,-1), (2,1) and (3,3)

We join the points up with a straight line.

**Example: Draw the graph for y = 3x + 1**

Here we have not been given a table of values.

We can create a table of values. We can see that the x values on the graph are -2, -1, 0, 1 and 2. We can use these for our table of values:

x |
-2 | -1 | 0 | 1 | 2 |

y |

We can now complete our table bu substituting our x values into the equation:

When x = 2

y = 3(2) + 1

y = 7

When x = 1

y = 3(1) + 1

y = 4

When x = 0

y = 3(0) + 1

y = 1

When x = -1

y = 3(-1) + 1

y = -2

When x = -2

y = 3(-2) + 1

y = -5

We can use these values to complete the table

x |
-2 | -1 | 0 | 1 | 2 |

y |
-5 | -2 | 1 | 4 | 7 |

The next step is to plot the coordinates:

(-2,-5), (-1,-2), (0,1), (1,4) and (2,7)

And finally we use a ruler to draw the line

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