4. Linear Functions

Lesson

There are a number of ways of stating an equation for a straight line. Previously, we saw slope-intercept form and we will see others in future lessons. Now, we'll discover the value of writing equations in standard form.

The standard form

The standard form of a linear equation is

$Ax+By=C$`A``x`+`B``y`=`C`

where $A$`A`, $B$`B`, and $C$`C` are all integers and the value of $A$`A` is **positive**, that is, $A>0$`A`>0.

When we are given an equation in standard form, we can either graph using intercepts or rearrange it to one of our other forms.

The standard form of a line is great for identifying both the $x$`x` and $y$`y` intercepts.

For example, the line $2x+3y=6$2`x`+3`y`=6

The $x$`x` intercept happens when the $y$`y` value is $0$0.

$2x+3y$2x+3y |
$=$= | $6$6 |

$2x+3\left(0\right)$2x+3(0) |
$=$= | $6$6 |

$2x$2x |
$=$= | $6$6 |

$x$x |
$=$= | $3$3 |

The $y$`y` intercept happens when the $x$`x` value is $0$0.

$2x+3y$2x+3y |
$=$= | $6$6 |

$2\left(0\right)+3y$2(0)+3y |
$=$= | $6$6 |

$3y$3y |
$=$= | $6$6 |

$y$y |
$=$= | $2$2 |

From here, we just need to plot the $x$`x` intercept $3$3, and the $y$`y` intercept $2$2, and draw the line through both.

If we don't want to graph using intercepts, we can also rearrange the equation to slope-intercept form. Let's look at rearranging between the two forms.

Express the equation $4x+6y=12$4`x`+6`y`=12 in slope-intercept form.

**Think:** We need to solve for $y$`y` to get the equation to the form $y=mx+b$`y`=`m``x`+`b`.

**Do:** To solve for $y$`y`, we need to reverse the operations to get $y$`y` by itself.

$4x+6y$4x+6y |
$=$= | $12$12 | Start with the given equation |

$6y$6y |
$=$= | $-4x+12$−4x+12 |
Undo addition by subtracting $4x$4x from both sides |

$y$y |
$=$= | $\frac{-4x+12}{6}$−4x+126 |
Undo multiplication by dividing by $6$6 on both sides |

$y$y |
$=$= | $\frac{-2}{3}x+2$−23x+2 |
Simplify |

**Reflect**: $y$`y` is isolated, so the linear equation $4x+6y=12$4`x`+6`y`=12 is $y=\frac{-2}{3}x+2$`y`=−23`x`+2 in slope-intercept form.

As a mathematical convention, we are often asked to give our answer in standard form. Let's look at rearranging to standard form.

Express the equation $y=\frac{-3}{4}x-5$`y`=−34`x`−5 in standard form.

**Think:** We need get to the form $Ax+By=C$`A``x`+`B``y`=`C`, where $A$`A`, $B$`B` and $C$`C` are integers and $A>0$`A`>0.

**Do:** We need to ensure there are no non-integer coefficients and that $A$`A` is positive.

$y$y |
$=$= | $\frac{-3}{4}x-5$−34x−5 |
Start with the given equation |

$4y$4y |
$=$= | $-3x-20$−3x−20 |
Clear the fraction by multiplying by $4$4 |

$3x+4y$3x+4y |
$=$= | $-20$−20 | Add $3x$3x to both sides to get the x and y terms to the same side |

**Reflect**: We now have something of the form $Ax+By=C$`A``x`+`B``y`=`C`, where $A$`A`, $B$`B` and $C$`C` are integers and$A>0$`A`>0, so the linear equation $y=\frac{-3}{4}x-5$`y`=−34`x`−5 is $3x+4y=-20$3`x`+4`y`=−20 in standard form.

Express the following equations in standard form.

$y=6x-5$

`y`=6`x`−5$y=\frac{6x}{5}-6$

`y`=6`x`5−6

Consider the line given by the equation: $5x-3y=-15$5`x`−3`y`=−15

Solve for $x$

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

`y`-value of the $y$`y`-intercept of the line.Hence, graph the equation of the line.

Loading Graph...

A line has slope $\frac{7}{3}$73 and passes through the point $\left(7,-9\right)$(7,−9).

By substituting into the equation $y=mx+b$

`y`=`m``x`+`b`, find the value of $b$`b`for this line.Write the equation of the line in standard form.

Create, interpret, and/or use the equation, graph, or table of a linear function.

Translate from one representation of a linear function to another (i.e., graph, table, and equation).

Write or identify a linear equation when given • the graph of the line Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

Write or identify a linear equation when given • two points on the line, or Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

Write or identify a linear equation when given • the slope and a point on the line. Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.