Solving Multi-Step Equations with Fractions Study Notes

Administrative and Course Details

• Course Identifier: Math 8 IB

• Publishing Organization: Kuta Software LLC (2019)

• Software Generator: Infinite Algebra 1

• Worksheet Document Identifier: ID: 1

• Core Curriculum Unit: Solving Multi-step equations with fractions

Methodological Framework for Solving Multi-Step Fractional Equations

To solve multi-step linear equations containing rational expressions, follow a systematic process to clear denominators and isolate the target variable:

1. Identify Denominators and Common Denominators:

   • Examine all fraction terms in the equation.

   • Determine either the Least Common Denominator (LCD) or any suitable Common Denominator (CD) across all denominators.

2. Clear Denominators (Multiplication Step):

   • Multiply every term on both sides of the equal sign by the chosen LCD or CD.

   • Simplify each term so that all coefficients and constants become integers.

3. Combine Like Terms:

   • Group like variable terms together on one side of the equation.

   • Group constant numerical values together on the appropriate side.

4. Isolate the Variable:

   • Perform inverse operations (addition/subtraction) to move constants away from the variable term.

   • Perform inverse operations (multiplication/division) to solve for the individual variable.

5. Simplify Rational Solutions:

   • Reduce fractions to their simplest improper or proper form.

Problem Solutions and Step-by-Step Analyses

Section 1: Equations with Denominators 15 and 30

Variable x System

• Common Denominator (CD): $30$

• Intermediate integer-coefficient equation: $-128 = 30x - 55 + 5x$

• Step 1: Combine variable terms on the right side: $30x + 5x = 35x$

• Step 2: Add constant $55$ to both sides: $-188 + 55 = 35x$ $-133 = 35x$

• Step 3: Solve for $x$: $x = -\frac{133}{35}$

Variable p System

• Least Common Denominator (LCD): $15$

• Intermediate equation after clearing denominators: $15p + 25 + 20p = 73$

• Step 1: Combine variable terms on the left side: $35p + 25 = 73$

• Step 2: Subtract $25$ from both sides: $35p = 73 - 25$ $35p = 48$

• Alternative intermediate forms and related calculations: $35p - 13 = 25$ $35p = 38$ $35p = -98$ $p = -\frac{98}{35}$

• Primary final exact form: $15p = 73$ $p = \frac{73}{15}$

Section 2: Problem 3 Solution

• Original equation: $-n + 5n = 4$

• Step 1: Combine like variable terms on the left-hand side: $4n = 4$

• Step 2: Divide both sides by $4$: $n = 1$

Section 3: Problem 4 Solution

• Original fractional equation: $\frac{1}{6}k + \frac{10}{6} + 2 = -\frac{11}{6}$

• Least Common Denominator (LCD): $6$

• Step 1: Multiply every term by $6$ to eliminate denominators: $6k + 13 + 12 = -11$

• Step 2: Combine constant terms on the left side and transpose: $6k = -11 - 13 - 12$ $6k = -36$

• Step 3: Divide both sides by $6$: $k = -6$

• Expanded verification step: $6k + 15 + 18 = -11 - 25$ $6k = -36$ $k = -6$

Section 4: Problem 5 Solution

Rational Equation with Variable b

• Least Common Denominator (LCD): $75$

• Primary equation with cleared denominators: $17 - 95b - 25 = 75$

• Step 1: Rearrange and group constant values on one side: $17 + 25 + 75 = 95b$ $342 = 95b$

• Step 2: Express $b$ as a simplified fraction: $b = \frac{342}{95}$

• Step 3: Divide numerator and denominator by their common factor $19$: $b = \frac{18}{5}$

Parallel System with Variable k

• Least Common Denominator (LCD): $6$

• Linear equation: $19 = 36k + 19 + 6$

• Step 1: Subtract constants from both sides: $19 - 19 - 6 = 6k$ $-6 = 6k$

• Step 2: Solve for $k$: $k = -1$

Section 5: Problem 18 Solution

System with Variable n

• Given equation: $\frac{19}{30} = \frac{1}{2}n + \frac{5}{9}n$

• Least Common Denominator (LCD): $90$

• Step 1: Multiply each term by $90$: $45n + 50n = 19$ $95n = 19$

• Step 2: Solve for $n$: $n = \frac{19}{95}$

• Step 3: Reduce fraction to lowest terms: $n = \frac{1}{5}$

System with Variable m

• Given equation: $\frac{7}{3}m + 2 - \frac{1}{3}m - 20 = \frac{7}{9}$

• Least Common Denominator (LCD): $90$

• Step 1: Clear denominators across all terms: $210m + 162 - 240 = 217$

• Step 2: Transpose constant values to the right side: $210m = 217 - 162 + 405$ $210m = 460$

• Step 3: Solve for $m$ and simplify: $m = \frac{460}{210}$ $m = \frac{23}{21}$

Section 6: Problem 10 Solution and Numerical Scratchwork

• Denominator Analysis:- Least Common Denominator (LCD): $30$

• Algebraic Transformations: $-46 - 15 - 45n$ $-46 - 61 = -107$ $61 = 46b$

• Multi-term rearrangement: $46b - 75 + 150 = 311$ $406b = 256$ $b = \frac{256}{406}$

• Decimal Long Division Calculations:- Primary quotient setup: $2160 / 506$

• Subtraction steps and remainder values: $418$, $186$, $5.106$, $1513$

• Resulting decimal approximation: $b \approx 6.4256$

Section 7: Practice Questions

1. Solve for $x$: $\frac{1}{2}x + \frac{1}{3}x = 10$

2. Solve for $y$: $\frac{3}{4}y - 2 = \frac{1}{2}y + 3$

3. Solve for $a$: $\frac{2}{5}a + \frac{1}{10} = \frac{1}{2}a - \frac{3}{10}$

4. Solve for $m$: $\frac{m}{3} + \frac{m}{4} = 7$

5. Solve for $k$: $\frac{5}{6}k - \frac{1}{3} = \frac{1}{2}k + \frac{2}{3}$

6. Solve for $p$: $\frac{2}{3}p + 4 = \frac{5}{9}p + 7$

7. Solve for $n$: $-2n + \frac{1}{2} = \frac{3}{4}n - 6$

8. Solve for $b$: $\frac{1}{5}b + \frac{2}{5}b + 3 = 12$

9. Solve for $z$: $\frac{3}{8}z - 1 = \frac{1}{4}z + 2$

10. Solve for $w$: $\frac{7}{12}w - \frac{1}{4} = \frac{1}{3}w + \frac{1}{2}$

11. Solve for $x$: $\frac{2}{7}x - \frac{5}{14}x = \frac{1}{2}$

12. Solve for $c$: $\frac{c+2}{4} = \frac{c-1}{3}$

13. Solve for $d$: $\frac{2d-1}{5} = \frac{d+3}{2}$

14. Solve for $v$: $\frac{1}{2}v + \frac{2}{3}v - \frac{1}{6}v = 8$

15. Solve for $t$: $\frac{4}{5}t - 3 = \frac{1}{2}t + 6$

16. Solve for $h$: $\frac{h}{6} + \frac{h}{9} = 5$

17. Solve for $r$: $\frac{3}{2}r - \frac{1}{4}r = \frac{5}{8}$

18. Solve for $q$: $\frac{q-3}{2} + \frac{q+1}{4} = 5$

19. Solve for $x$: $\frac{5}{3}x - 4 = \frac{2}{3}x + 2$

20. Solve for $g$: $\frac{3g}{10} - \frac{g}{5} = \frac{1}{2}$