4. Linear Equations, Proportional Relationships & Lines

Lesson

The slope (sometimes called slope) is the rate of change over an interval or the ratio of vertical change to horizontal change.

$\text{slope }=\frac{\text{vertical change }}{\text{horizontal change }}$slope =vertical change horizontal change

We refer to the horizontal measurement as the *run *and the vertical measurement as the *rise*. This gives us the more common rise over run formula.

Slope formula

$\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run

To find the slope of the line shown above we will count how many units it rises (the vertical distance) and how many it runs (the horizontal distance). In this case the $rise=4$`r``i``s``e`=4 units and the $run=4$`r``u``n`=4 units as well. So the $\frac{rise}{run}=\frac{4}{4}$`r``i``s``e``r``u``n`=44 which simplifies to a slope of $1$1.

Find the slope of the line graphed below.

**Think:** We need to find the rise and run but here should we start counting from and where should we stop?

**Do:** Find to *nice* points. These will be points that lie on the line where the grid lines intersect. By choosing these points we are making sure to avoid decimal or fractional values for the rise and run. The points $\left(-3,-2\right)$(−3,−2) and $\left(0,2\right)$(0,2) will work nicely here. but if there were any other nice points visible on the graph we could use whichever ones we wanted to and still get the same answer. When that is the case it is usually best to avoid points with negative coordinates to make sure we don't make any errors in our calculations.

By labeling the vertical and horizontal change we can see that the $rise=4$`r``i``s``e`=4 and the $run=3$`r``u``n`=3 so the slope is $\frac{rise}{run}=\frac{4}{3}$`r``i``s``e``r``u``n`=43. Since this fraction doesn't reduce we can keep the slope as $\frac{4}{3}$43.

**Reflect:** By looking at the graph we can easily see that the slope should be positive because it increases from left to right (or goes uphill) so we know we haven't made any sign errors when calculating the rise and run.

It is always nice to have a graph when we are finding the slope of a line, but we don't necessarily need one. Let's start with a small table that only has two entries.

x | y |
---|---|

3 | 6 |

7 | -2 |

Remember that a table is just another way of organizing ordered pairs. This means that this table represents the ordered pairs $\left(3,6\right)$(3,6) and $\left(7,-2\right)$(7,−2). If we want, we can plot these points on a graph and find the slope using $\frac{\text{rise }}{\text{run }}$rise run .

Looking at the graph, the $rise=-8$`r``i``s``e`=−8. It is negative because even though we call it rise, we have to move down to get from point $A$`A` to point $B$`B`. The $run=4$`r``u``n`=4. So the slope is:

$\frac{\text{rise }}{\text{run }}=\frac{-8}{4}$rise run =−84 $=$= $-2$−2

This is a great way to find the slope, but we may not want to make a graph for every table. We don't need a visual to be able to find the vertical and horizontal distances between points $A$`A` and $B$`B`. Remember that the $x$`x`-coordinate of a point represents its horizontal position on the graph and the $y$`y`-coordinate represents its vertical position on the graph. So if we can find the difference between the $x$`x`-coordinates and the $y$`y`-coordinates of points $A$`A` and $B$`B`, then we can find the rise and run without a graph.

Remember to find the difference between two numbers we subtract them. Subtracting the $y$`y`-coordinates we find the rise is:

$6-\left(-2\right)=8$6−(−2)=8

Notice the sign is different from what we got before. Let's see if that changes our final answer.

Subtracting the $x$`x`-coordinates we find the run is:

$3-7=-4$3−7=−4

This sign is also different from what we got before.

Using these new values we get a slope of:

$\frac{\text{rise }}{\text{run }}=\frac{8}{-4}$rise run =8−4 $=$= $-2$−2

Determine the slope from the table below.

x | y |
---|---|

-2 | 4 |

-1 | 7 |

0 | 10 |

1 | 13 |

**Think:** Recall that this table actually represents a list of ordered pairs. We could graph them or use $\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run .

**Do: **

Remember rise represents the change in $y$`y` values, to find the change in $y$`y` values we can subtract them. To find the run, we can subtract the $x$`x` values.

$\text{slope }$slope | $=$= | $\frac{\text{rise }}{\text{run }}$rise run |

$=$= | $\frac{-2-6}{7-3}$−2−67−3 | |

$=$= | $\frac{-8}{4}$−84 | |

$=$= | $-2$−2 |

Notice, we subtracted the $x$`x` values and the $y$`y` values in the same order.

**Reflect:** Would it answer change if we switched the order of the subtraction and did $\frac{6-\left(-2\right)}{3-7}$6−(−2)3−7? Try it.

Careful!

When finding the slope, the most common error is when students are not consistent with the order in which they subtract. If we put the second $y$`y` value at the begging of the subtraction for the rise then we must put the second $x$`x` value at the beginning of the subtraction for the run.

What is the slope of the line shown in the graph, given that Point A $\left(3,3\right)$(3,3) and Point B $\left(6,5\right)$(6,5) both lie on the line?

Loading Graph...

Find the slope of the line that passes through Point A $\left(3,5\right)$(3,5) and Point B $\left(1,8\right)$(1,8), using $m=\frac{y_2-y_1}{x_2-x_1}$`m`=`y`2−`y`1`x`2−`x`1.

The $y$`y`-intercept is the point where a graph crosses the $y$`y`-axis. The $y$`y`-intercept of the graph shown below is $\left(0,2\right)$(0,2).

What is the $y$`y`-intercept of the graph shown below?

**Think: **Where does the line cross the $y$`y`-axis? What are the coordinates of that point?

**Do:** The point where the line crosses the $y$`y`-axis is $\left(0,1\right)$(0,1). That is the $y$`y`-intercept.

**Reflect:** Is there anything special about the $x$`x`-coordinate of every $y$`y`-intercept? Does this graph have any other types of intercepts?

Notice from the examples above that the $x$`x`-coordinate of every $y$`y`-intercept is $0$0. This is because if a point is located on the $y$`y`-axis then it is always located at $0$0 on the $x$`x`-axis. So if we are looking at a table, we simply need to find $0$0 in the $x$`x` column. The associated $y$`y` value will give us our $y$`y`-intercept. What is the $y$`y`-intercept of the table shown below?

x | y |
---|---|

-2 | 4 |

-1 | 7 |

0 | 10 |

1 | 13 |

If we find $0$0 in the $x$`x` column we see that the associated $y$`y` value is $10$10. This means the $y$`y`-intercept is $\left(0,10\right)$(0,10), but we noticed that the $x$`x`-coordinate is always $0$0 for the $y$`y`-intercept so we can just say that the $y$`y`-intercept is $10$10 because it is implied that the $x$`x` value is $0$0.

We have looked at this graph previously and discovered that the slope is $\frac{4}{3}$43 and the $y$`y`-intercept is $2$2 or $\left(0,2\right)$(0,2).

The equation of this line is $y=\frac{4}{3}x+2$`y`=43`x`+2. We can see that the slope has become the coefficient on the variable $x$`x` and the $y$`y`-intercept has become the constant that is added at the end. This is called the slope-intercept form of a line because from the equation we can easily identify the slope and the $y$`y`-intercept.

Slope-intercept form of a line

The general slope-intercept form of a line, where $m$`m` is the slope and $b$`b` is the $y$`y`-intercept is:

$y=mx+b$`y`=`m``x`+`b`

Given the equation $y=3-2x$`y`=3−2`x`, identify the slope and $y$`y`-intercept.

**Think:** This equation isn't quite in the form we are used to. Can we rewrite it so that it takes the form $y=mx+b$`y`=`m``x`+`b`?

**Do: **Rewrite the equation as $y=-2x+3$`y`=−2`x`+3. Now we can see the coefficient on the $x$`x` and we can see that $3$3 is the constant term. Using $y=mx+b$`y`=`m``x`+`b` as a reference, we can see that $m=-2$`m`=−2 and $b=3$`b`=3. So the slope is $-2$−2 and the $y$`y`-intercept is $3$3.

Consider the equation $y=x$`y`=`x`. A table of values is given below.

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

$y$y |
$-1$−1 | $0$0 | $1$1 | $2$2 |

Plot the points in the table of values.

Loading Graph...Find the coordinates of the $y$

`y`-intercept.Find the slope of the line.

State the slope and $y$`y`-intercept of the equation $y=-4x$`y`=−4`x`.

**Slope**$\editable{}$ **$y$**`y`-intercept$\editable{}$