Solving multi-step linear equations with fractions requires a clear understanding of algebraic principles and fraction operations. This process involves systematically isolating the variable by performing operations such as addition, subtraction, multiplication, and division on both sides of the equation.

In this article, we’ll break down the process of solving multi-step linear equations with fractions into simple, manageable steps. My goal is to make sure you feel confident and capable when tackling these problems. We’ll cover the essential techniques, highlight common mistakes to avoid, and provide practical examples to show how these equations are used in real life. Plus, we’ll include FAQs and solved problems to help reinforce your understanding.

What is Multi-Step Linear Equations with Fractions?

Multi-step linear equations with fractions are algebraic equations that involve multiple operations (addition, subtraction, multiplication, division) to isolate the variable. These equations also contain fractions, which add an extra layer of complexity. The goal is to simplify the equation step-by-step until the variable is isolated on one side of the equation.

• Fractions: Parts of a whole, represented as a ratio of two integers, such as [Tex]\frac{a}{b}[/Tex], Where a is the numerator and b numerator and b is the denominator.
• Operations: The sequence of mathematical actions performed to simplify and solve the equation, including addition, subtraction, multiplication, and division.
• Isolation of the Variable: The process of manipulating the equation to get the variable (usually represented as x ) alone on one side.

Example Multi-Step Linear Equations with Fractions

Consider the equation [Tex]\frac{3x}{5} – \frac{2}{3} = \frac{1}{2}[/Tex]. To solve this equation, follow these steps :

Step 1: Find the least common denominator (LCD) of fractions, i.e. 30.

Step 2: Multiply each term by 30 to clear the fractions:

30(3x/5) – 30(2/3) = 30(1/2)

Step 3: Simplify: 18x – 20 = 15

Step 4: Isolate the variable x by adding 20 to both sides: 18x = 35

Step 5: Finally, divide by 18 to solve for x: x = 35/18

Important Formulas Multi-Step Linear Equations with Fractions

Understanding key formulas and concepts is crucial for solving multi-step linear equations with fractions. Here are the important ones:

Distributive Property

The distributive property allows you to eliminate parentheses by distributing multiplication over addition or subtraction within the parentheses.

• Formula: a(b+c) = ab + ac

Combining Like Terms

Combine terms with the same variable to simplify the equation.

• Formula: ax + bx = (a+b)x

Addition and Subtraction Properties of Equality

Add or subtract the same value from both sides of the equation to help isolate the variable.

• Formula: if a = b, than a + c = b+c and a-c = b-c

Multiplication and Division Properties of Equality

Multiply or divide both sides of the equation by the same nonzero number to isolate the variable.

• Formula: If a=b , than a*c = b*c (for [Tex]c \neq 0[/Tex]) and [Tex]\frac{a}{c} = \frac{b}{c} [/Tex](for [Tex]c \neq 0[/Tex]).

Reciprocal of a Fraction

When dividing by a fraction, multiply by its reciprocal.

• Formula: [Tex]\frac{a}{\frac{b}{c}} = a\cdot \frac{c}{b}[/Tex]

Cross-Multiplication (for Equations with Two Fractions)

For equations involving two fractions set equal to each other, cross-multiply to clear the fractions.

• Formula: If [Tex]\frac{a}{b} = \frac{c}{d},[/Tex] then ad = bc.

Linear Equations with Fractions

Concept related to Linear equation with fractions is added below:

Definition: Linear equations are mathematical statements of equality involving variables and constants. When fractions are included, these equations can be written in the form , where a,b, c,d, e are integers, and c and e are not zero.

Examples of Linear Equations with Fractions
[Tex]\bold{\frac{3x+2}{4} = \frac{5}{6}}[/Tex]
[Tex]\bold{\frac{2x-7}{5} + \frac{3}{8} = \frac{4}{9}}[/Tex]

Steps to Solve Multi-Step Linear Equations with Fractions

Follow the steps added below to Solve Multi-Step Linear Equations with Fractions

Step 1: Simplify the Fractions

Ensure all fractions are in their simplest form. For example, reduce [Tex]\frac{4}{8}[/Tex] to [Tex]\frac{1}{2}[/Tex]

Step 2: Find the Least Common Denominator (LCD)

Identify the LCD of all the fractions in the equation. The LCD is the smallest number that all denominators can divide into evenly.

Step 3: Clear the Fractions

Multiply every term in the equation by the LCD to eliminate the fractions. For example, in the equation [Tex]\bold{\frac{2x}{3} + \frac{1}{4} = \frac{5}{6}}[/Tex], the LCD is 12. Multiply every term by 12:
[Tex]\bold{12(\frac{2x}{3}) + 12(\frac{1}{4}) = 12(\frac{5}{6})} [/Tex]
[Tex]\bold{8x + 3 = 10}[/Tex]

Step 4: Solve the Simplified Equation

Solve the resulting linear equation using standard algebraic methods. For example:
[Tex]\bold{8x + 3 = 10}[/Tex]
[Tex]\bold{8x = 7}[/Tex]
[Tex]\bold{x = \frac{7}{8}}[/Tex]

Common Mistakes and How to Avoid Them

• Misinterpreting Fractions : Always double-check the fractions to ensure they are correctly understood.

• Incorrectly Simplifying Fractions : Verify that all fractions are simplified before proceeding.

• Errors in Arithmetic Operations : Carefully perform all arithmetic operations and recheck your work to avoid mistakes.

Examples Related to Solving Multi-Step Linear Equations with Fractions

Example 1: Solve [Tex]\bold{\frac{3x}{5} – \frac{2}{3} = \frac{1}{2}}[/Tex]

Solution:

Find the Least Common Denominator (LCD) of 5, 3, and 2, which is equal to 30.

Multiply through by 30:
[Tex]\bold{30(\frac{3x}{5}) – 30(\frac{2}{3}) = 30(\frac{1}{2})} [/Tex]
[Tex]\bold{18x − 20 = 15} [/Tex]
[Tex]\bold{18x=35}[/Tex]
[Tex]\bold{x = \frac{35}{18}}[/Tex]

Example 2: Solve [Tex]\bold{\frac{4x}{7} + \frac{5}{6} = \frac{3}{2}}[/Tex]

Solution:

Find the LCD of 7, 6, and 2, which is equal to 42.

Multiply through by 42:
[Tex]\bold{42(\frac{4x}{7}) + 42(\frac{5}{6}) = 42(\frac{3}{2})} [/Tex]
[Tex]\bold{24x+35=63}[/Tex]
[Tex]\bold{24x=28}[/Tex]
[Tex]\bold{x = \frac{28}{24} = \frac{7}{6}}[/Tex]

Example 3: Solve [Tex]\frac{2x}{3} + \frac{1}{4} = \frac{5}{6}[/Tex]

Solution:

Find Least Common Denominator (LCD): The LCD of 3, 4, and 6 is 12

Multiply every term by 12:

[Tex]12 \cdot \frac{2x}{3} + 12 \cdot \frac{1}{4} = 12 \cdot \frac{5}{6}[/Tex]

Simplifies to:

8x + 3 = 10

Subtract 3 from both sides:

8x = 7

x = 7/8

Example 4: Solve [Tex]\frac{4x}{7} – \frac{3}{5} = \frac{1}{2}[/Tex]

Solution:

LCD of 7, 5, and 2 is 70.

Multiply every term by 70

[Tex]70 \cdot \frac{4x}{7} – 70 \cdot \frac{3}{5} = 70 \cdot \frac{1}{2}[/Tex]

Simplifies to:

40x – 42 = 35

Add 42 to both sides:

40x = 77

x = 77/40

Example 5: Solve [Tex]\frac{2x-3}{5} = \frac{4}{3}[/Tex]

Solution:

Cross-Multiply:

[Tex]3(2x-3) = 4 \cdot 5[/Tex]

Simplifies to:

6x – 9 = 20

Add 9 to both sides

6x = 29

x = 29/6

Example 6: Solve [Tex]\frac{7x-1}{3} = \frac{2x+5}{4}[/Tex]

Solution:

Cross-Multiply:

[Tex]4(7x – 1) = 3(2x+5)[/Tex]

Simplifies to:

28x – 4 = 6x + 15

Combine Like Terms: Subtract 6x from both sides:

22x – 4 = 15

Add 4 to both sides:

22x = 19

x = 19/22

Example 7: Solve [Tex]\frac{3x+2}{4} = \frac{5}{6}[/Tex]

Solution:

Cross-Multiply:

[Tex]6(3x + 2) = 5 \cdot 4[/Tex]

Simplifies to:

18x + 12 = 20

Subtract 12 from both sides:

18x = 8

x = 8/18 = 4/9

Example 8: Solve [Tex]\frac{6x+1}{2} = \frac{3x-4}{4}[/Tex]

Solution:

Cross-Multiply:

[Tex]4(6x+1) = 2(3x-4)[/Tex]

Simplifies to:

24x+4 = 6x-8

Combine Like Terms: Subtract 6x from both side:

18x+4 = -8

Subtract 4 from both sides:

18x = -12

x = -12/18 = -2/3

Example 9: Solve [Tex]\frac{3x-2}{5} = \frac{7}{3}[/Tex]

Solution:

Cross-Multiply:

[Tex]3(3x-2) = 7\cdot5[/Tex]

Simplifies to:

9x-6 = 35

Add 6 to both sides:

9x = 41

x = 41/9

Example 10: Solve [Tex]\frac{4x+3}{2} = \frac{5x-1}{3}[/Tex]

Solution:

Cross-Multiply:

[Tex]3(4x + 3) = 2(5x – 1)[/Tex]

Simplifies to:

12x + 9 = 10x -2

Combine Like Terms: Subtract 10x from both side :

2x + 9 = -11

Subtract 9 from both sides:

2x = -11

x = -11/2

Multi-Step Linear Equations with Fractions – Practice Problems

Problem 1: Solve: [Tex]\frac{5x-3}{7} = \frac{2x+4}{3}[/Tex]

Problem 2: Solve: [Tex]\frac{2x+1}{5} = \frac{x-3}{2}[/Tex]

Problem 3: Solve: [Tex]\frac{3x-2}{4} = \frac{x-3}{2}[/Tex]

Problem 4: Solve: [Tex]\frac{4x+3}{8} = \frac{5x-2}{7}[/Tex]

Problem 5: Solve: [Tex]\frac{7x-5}{9} = \frac{2x+6}{7}[/Tex]

Problem 6: Solve: [Tex]\frac{6x+2}{3} = \frac{4x-1}{5}[/Tex]

Problem 7:Solve: [Tex]\frac{5x-4}{6} = \frac{3x+2}{8}[/Tex]

Problem 8: Solve: [Tex]\frac{8x+1}{7} = \frac{2x-3}{5}[/Tex]

Problem 9: Solve: [Tex]\frac{9x-2}{4} = \frac{3x+5}{6}[/Tex]

Problem 10: Solve: [Tex]\frac{10x+3}{5} = \frac{4x-7}{9}[/Tex]

Conclusion

Understanding and applying the steps to solve these equations not only enhances your mathematical skills but also prepares you for practical problem-solving in numerous fields.

In this article, we’ve broken down the process into clear, manageable steps, highlighted common mistakes to avoid, and provided practical examples to demonstrate real-world applications. By practicing these methods and using the tips provided, you can approach multi-step linear equations with confidence and accuracy.

Read More:

• Simplifying Fractions
• Least Common Denominator (LCD)
• Solving Linear Equations
• Linear Equations in One Variable

Frequently Aksed Questions

Why is the use of Solving Linear Equations?

Solving fractions simplifies the equation, making it easier to solve.

What is the Least Common Denominator (LCD)?

LCD is the smallest number that all denominators in the equation can divide into evenly.

What are Simple Fractions?

Simple fractions, often just called “fractions,” are a way of representing parts of a whole or a division of quantities.

What are Examples of Simple Fractions?

Various examples of simple fractions are: 1/2, 3/4, 7/10, etc.