Linear Equations II

Hong Kong

Stage 1 - Stage 3

Lesson

Straight lines are lines on the Cartesian Plane that extend forever in both directions. If we ignore for a moment the special cases of horizontal and vertical lines, straight lines will cross both the $x$`x` -axis and the $y$`y`-axis or maybe through the point where the $x$`x` and $y$`y` axes intersect (called the origin).

Here are some examples...

The word **intercept **in mathematics refers to a point where a line, curve or function crosses or intersects with the axes.

- We can have $x$
`x`intercepts: where the line, curve or function crosses the $x$`x`axis. - We can have $y$
`y`intercepts: where the line, curve or function crosses the $y$`y`axis.

Consider what happens as a point moves up or down along the $y$`y`-axis. It will eventually reach the origin $\left(0,0\right)$(0,0) where $y=0$`y`=0. Now, if the point moves along the $x$`x`-axis in either direction, the $y$`y` value is still $0$0.

Similarly, consider what happens as a point moves along the $x$`x`-axis. It will eventually reach the origin where $x=0$`x`=0. Now, if the point moves along the $y$`y`-axis in either direction, the $x$`x` value is still $0$0.

This interactive demonstrates the idea behind the coordinates of x and y intercepts.

So, two important properties are:

- any point on the $x$
`x`-axis will have $y$`y`value of $0$0 - any point on the $y$
`y`-axis will have $x$`x`value of $0$0

We can use these properties to calculate or identify $x$`x` and $y$`y` intercepts for any line, curve or function.

Intercepts

The $x$`x` intercept occurs at the point where $y=0$`y`=0.

The $y$`y` intercept occurs at the point where $x=0$`x`=0.

Find the $x$`x` and $y$`y` intercepts for the following lines.

$y=3x$`y`=3`x`

**Think**: The $x$`x` intercept occurs when $y=0$`y`=0. The $y$`y` intercept occurs when $x=0$`x`=0.

**Do**: When $x=0$`x`=0, $y=3\times0$`y`=3×0= $0$0

This means that this line passes through $\left(0,0\right)$(0,0), the origin. The $x$`x` and $y$`y` intercept occur at the same point!

*This particular form of a straight line *$y=mx$`y`=`m``x` *always passes through the origin. (Test it out on the applet below) *

$y=4x-7$`y`=4`x`−7

**Think**: The $x$`x` intercept occurs when $y=0$`y`=0. The $y$`y` intercept occurs when $x=0$`x`=0.

**Do**:

When $x=0$`x`=0

$y=4\times0-7$`y`=4×0−7 = $-7$−7 So the $y$`y` intercept is $-7$−7

When

$y$y |
$=$= | $0$0 |

$0$0 | $=$= | $4x-7$4x−7 |

$4x$4x |
$=$= | $7$7 |

$x$x |
$=$= | $\frac{7}{4}$74 |

So the $x$`x` intercept is $\frac{7}{4}$74

*This form of a straight line* $y=mx+b$`y`=`m``x`+`b`, *always has* $y$`y` *intercept of* $b$`b`.

*The $x$ x intercept is easy to work out after that (substitute* $y=0$

$2y-5x-10=0$2`y`−5`x`−10=0

**Think**: The $x$`x` intercept occurs when $y=0$`y`=0. The $y$`y` intercept occurs when $x=0$`x`=0.

**Do**: When $x=0$`x`=0, the $5x$5`x` term 'disappears' as $5\times0=0$5×0=0. This leaves us with:

2*y-10 | $=$= | 0 |

2*y | $=$= | 10 |

y | $=$= | 5 |

So the $y$`y` intercept is $5$5

When $y=0$`y`=0, the $2y$2`y` term disappears. This leaves us with:

-5*x-10 | $=$= | 0 |

-5*x | $=$= | 10 |

x | $=$= | -2 |

So the $x$`x` intercept is $-2$−2

Here are some helpful hints when thinking about linear relationships:

A relationship is linear if the resulting graph is a straight line. So you can tell it if it is linear by plotting the graph. Another way to see if the relationship is linear is from looking at the equation. Linear equations have x variables in linear form (without powers). Here are some examples of linear equations:

$y=x$`y`=`x`

$y=2x+5$`y`=2`x`+5

$y=-\frac{1}{2}x-17$`y`=−12`x`−17

$y+x+4=0$`y`+`x`+4=0

$x=4$`x`=4

$x$`x`-intercepts occur when the $y$`y`-value is $0$0. So set $y=0$`y`=0 and then solve for $x$`x`. Alternatively read the $x$`x`-intercept value from a graph of the line.

$y$`y`-intercepts occur when the $x$`x`-value is $0$0. So set $x=0$`x`=0 and then solve for $y$`y`. Alternatively read the $y$`y`-intercept value from a graph of the line.

Finding other $x$`x` and $y$`y` values from a graph is similar to reading coordinates off a map.

To find a $y$`y`-value - start with the $x$`x`-value given, then travel vertically until you hit the line. Read the $y$`y`-value for this height off the axis.

To find a $x$`x`-value - start with the $y$`y`-value given, then travel horizontally until you hit the line. Read the $x$`x`-value for this point off the axis.

This interactive allows you to move a point around on a line and see how the coordinates change.

To graph any linear relationship you only need 2 points that are on the line. You can use any two points from a table of values, or substitute in any 2 values of x into the equation and solve for corresponding y-values to create your own two points. Often, using the intercepts is one of the easiest ways to sketch the line.

x | 1 | 2 | 3 | 4 |

y | 3 | 5 | 7 | 9 |

To sketch from a table of values, we need just any two points from the table. From this table we have 4 coordinates, $\left(1,3\right)$(1,3), $\left(2,5\right)$(2,5), $\left(3,7\right)$(3,7), $\left(4,9\right)$(4,9).

Drag the $2$2 of the points on this interactive to the correct positions for any 2 of the points above and graph this linear relationship.

If we are given the equation of a linear relationship, like $y=3x+5$`y`=3`x`+5, then to sketch it we need two points. We can pick any two points we like.

Start by picking any two $x$`x`-values you like, often the $x$`x`-value of $0$0 is a good one to pick because the calculation for y can be quite simple. For our example, $y=3x+5$`y`=3`x`+5 becomes $y=0+5$`y`=0+5, $y=5$`y`=5. This gives us the point $\left(0,5\right)$(0,5)

Similarly look for other easy values to calculate such as $1$1, $10$10, $2$2. I'll pick $x=1$`x`=1. Then for $y=3x+5$`y`=3`x`+5, we have $y=3\times1+5$`y`=3×1+5, $y=8$`y`=8.This gives us the point $\left(1,8\right)$(1,8)

Now we plot the two points and create a line.

It's always a nice idea to check your line using a third point. For this example I might pick the value $x=-1$`x`=−1.

$y=3\times\left(-1\right)+5$`y`=3×(−1)+5

$y=-3+5$`y`=−3+5

$y=2$`y`=2 and I can see that this point would lie on the line I have drawn.

Consider the equation $y=-2x$`y`=−2`x`.

Find the $y$

`y`-value of the $y$`y`-intercept of the line.Find the $x$

`x`-value of the $x$`x`-intercept of the line.Find the value of $y$

`y`when $x=2$`x`=2.Plot the equation of the line below.

Loading Graph...

Line L1 has the following equation: $y=x-2$`y`=`x`−2

Find the $y$

`y`value of the $y$`y`-intercept of the line.Find the $x$

`x`value of the $x$`x`-intercept of the line.Find the $y$

`y`-coordinate of a point that has an $x$`x`-coordinate of $-5$−5.Plot the line $y=x-2$

`y`=`x`−2 on the number plane.Loading Graph...

Consider the line with equation: $3x+y+2=0$3`x`+`y`+2=0

Solve for the $x$

`x`-value of the $x$`x`-intercept of the line.Solve for the $y$

`y`-value of the $y$`y`-intercept of the line.Plot the line.

Loading Graph...