Linear Equations

Lesson

A point on the $xy$`x``y`-plane represents a pair of quantities, the $x$`x`-value and the $y$`y`-value. We can write this pair in the form $\left(x,y\right)$(`x`,`y`), which we call an ordered pair. We say that a set of points on the $xy$`x``y`-plane forms a linear relationship if we can pass a single straight line that goes through all the points.

Each column in a table of values may be grouped together in the form $\left(x,y\right)$(`x`,`y`), which we know as an ordered pair. Let's consider the following table of values:

$x$x |
$1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|

$y$y |
$-2$−2 | $1$1 | $4$4 | $7$7 |

The table of values has the following ordered pairs:

$\left(1,-2\right),\left(2,1\right),\left(3,4\right),\left(4,7\right)$(1,−2),(2,1),(3,4),(4,7)

We can plot each ordered pair as a point on the $xy$`x``y`-plane.

Points plotted from the table of values |

We can plot the ordered pair $\left(a,b\right)$(`a`,`b`) by first identifying where $x=a$`x`=`a` along the $x$`x`-axis and $y=b$`y`=`b` along the $y$`y`-axis.

Take $\left(3,4\right)$(3,4) as an example. We first identify $x=3$`x`=3 along the $x$`x`-axis and draw a vertical line through this point. Then we identify $y=4$`y`=4 along the $y$`y`-axis and draw a horizontal line through that point. Finally we plot a point where the two lines meet, and this represents the ordered pair $\left(3,4\right)$(3,4).

In the example above, we can draw a straight line that passes through these points like so:

Straight line passing through all four points |

So we say that these points form a linear relationship.

We may also be interested in finding out whether an **additional** point also lies on the straight line that passes through the other points. Consider the ordered pair $\left(0,-5\right)$(0,−5). If we plot this ordered pair on the $xy$`x``y`-plane, then we obtain the following:

Straight line passing through additional point |

So, we can say that the additional point lies on the line or that all five points form a linear relationship.

Alternatively, we can refer to the table of values to determine whether the ordered pair $\left(0,-5\right)$(0,−5) lies on the line. Let's refer back to the table of values from before, but consider the additional point when $x=0$`x`=0:

$x$x |
$0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|

$y$y |
$\editable{}$ | $-2$−2 | $1$1 | $4$4 | $7$7 |

The value of $y$`y` decreases by $3$3 as the value of $x$`x` decreases by $1$1. So we expect to find $y=-5$`y`=−5 when $x=0$`x`=0. This is exactly the ordered pair $\left(0,-5\right)$(0,−5), which tells us that this point satisfies the linear relationship between $x$`x` and $y$`y`.

Do the points on the plane form a linear relationship?

Loading Graph...

Yes

ANo

B

Consider the table of values given below.

$x$x |
$-2$−2 | $-1$−1 | $0$0 | $1$1 |
---|---|---|---|---|

$y$y |
$6$6 | $2$2 | $-2$−2 | $-6$−6 |

Plot the points in the table of values.

Loading Graph...Is the relationship in the table of values linear?

No

AYes

B

Consider the table of values given below.

$x$x |
$-2$−2 | $-1$−1 | $0$0 | $1$1 |
---|---|---|---|---|

$y$y |
$2$2 | $-1$−1 | $-4$−4 | $-7$−7 |

Does the ordered pair $\left(-4,8\right)$(−4,8) satisfy the linear relationship between $x$

`x`and $y$`y`?No

AYes

B