4. Linear Equations, Proportional Relationships & Lines

Lesson

Here we are looking at solving multi-step equations that include fractions. We can follow the exact same steps that we did when solving equations with whole numbers, after we simply the equation.

To simplify an equation that has fractional terms, multiply both sides of the equation by the least common multiple of all fractions that appear in the equation. If you have correctly identified the least common denominator of all of the fractions, this will result in multiplying a whole number each of the fractional terms that will cancel out the denominator of all of the fractions.

Once again we will follow the reversed order of operations in order to isolate the variable.

Solve the following equation $\frac{n}{2}+\frac{n}{3}=\frac{15}{4}$`n`2+`n`3=154.

**Think:** What is the least common multiple (LCM) of the denominators of the fractions in this equation? The LCM of $2$2 ,$3$3 and $4$4 is $12$12. So, to remove the fractions from this equation, we will multiply both sides of the equation by $12$12.

**Do: **

$\frac{n}{2}+\frac{n}{3}$n2+n3 |
$=$= | $\frac{15}{4}$154 |
Notice the LCD of the denominators is $12$12 |

$12\left(\frac{n}{2}+\frac{n}{3}\right)$12(n2+n3) |
$=$= | $12\left(\frac{15}{4}\right)$12(154) |
Multiply both sides of the equation by $12$12 |

$\frac{12}{1}\times\frac{n}{2}+\frac{12}{1}\times\frac{n}{3}$121×n2+121×n3 |
$=$= | $\frac{12}{1}\times\frac{15}{4}$121×154 |
Be sure that you have distributed the $12$12 to each term on both sides of the equation |

$6n+4n$6n+4n |
$=$= | $45$45 |
If you chose the correct LCM, the denominators of all of the fractions will cancel out |

$10n$10n |
$=$= | $45$45 |
Combine like terms |

$\frac{10n}{10}$10n10 |
$=$= | $\frac{45}{10}$4510 |
Divide both sides by $10$10 to isolate the variable |

$n$n |
$=$= | $4.5$4.5 |

**Reflect: **If the choice for least common multiple of the denominators of all of the fractions in the equation is correct, then when both sides of the equation are multiplied by $12$12 it should remove all of the fractions in the equation. As seen in the third and fourth rows of the work above, this had the result of eliminating all of the fractions in the equation - making it much simpler to solve the equation.

Here are some more multi-step equations that the same technique may be applied to for you to try.

Solve the following equation: $\frac{9x}{3}+\frac{9x}{2}=-5$9`x`3+9`x`2=−5

Solve the following equation: $\frac{5x}{3}-3=\frac{3x}{8}$5`x`3−3=3`x`8

Solve the following equation: $\frac{8x-2}{3}=\frac{6x-3}{4}$8`x`−23=6`x`−34

Solve linear equations with rational number coefficients fluently, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Represent real-world problems using linear equations and inequalities in one variable and solve such problems.