Equations

Lesson

We have already seen how to solve Single Step Equations. These ones seemed almost too simple to use a series of rules or steps to solve. Time to take it to the next level.

A two step equation is one that will require two steps to solve. They generally have a multiplication/division and a subtraction/addition. The following are all two step equations.

$2x+5=11$2`x`+5=11, $\frac{1}{3}h-3=6$13`h`−3=6, $19-2j=5$19−2`j`=5

Let's look at two methods for solving two step equations.

Two step equations can be set up using a backtracking tool.

Start by writing the variable (variable) in a box. Then one step at a time mark in the operations that happen in order (according to Order of Operations). Remember we learnt how to set up equations in backtracking here.

Solve: $-2x+4=8$−2`x`+4=8

**Think**: First we need to set up the equation.

× $-2$−2 | $+$+$4$4 | |||

$x$x |
$8$8 |

**Do**: Backtrack one step at a time, reversing each operation.

× $-2$−2 | $+$+$4$4 | |||

$x$x |
$4$4 | $8$8 |

× $-2$−2 | $+$+$4$4 | |||

$-2$−2 | $4$4 | $8$8 |

So $x=-2$`x`=−2.

With all equations we can check our solution. Does $x=-2$`x`=−2 satisfy the equation $-2x+4=8$−2`x`+4=8?

This means using the "do the same to both sides" method, to isolate the $x$`x`. That is, get the $x$`x` on its own.

It is all about reversing the operations. So, we will need to remove the constant (number) term first. This is done by choosing the reverse operation.

$-2x+4=8$−2`x`+4=8

$-2x+4$−2x+4 |
$=$= | $8$8 | The opposite of addition is subtraction. |

$-2x+4-4$−2x+4−4 |
$=$= | $8-4$8−4 | Start by subtracting $4$4 from both sides. |

$-2x$−2x |
$=$= | $4$4 | Simplify both sides of the equation |

$\frac{-2x}{-2}$−2x−2 |
$=$= | $\frac{4}{-2}$4−2 | Then identify the next operation that needs to be reversed. The opposite of multiplying by $-2$−2 is dividing by $-2$−2. |

$x$x |
$=$= | $-2$−2 | Simplify both sides to find $x$x |

Solve the following equation: $8m+9=65$8`m`+9=65

Find the solution for the following equation: $\frac{x+9}{7}=4$`x`+97=4

Solve the following equation: $5\left(y+1\right)=25$5(`y`+1)=25