Fractions in equations can be tricky, but they're not impossible to handle. We'll learn how to check if a fraction works in an equation and how to solve equations with fractions. These skills are super useful for tackling more complex math problems.

We'll also explore how to turn word problems into fraction equations and solve them. This helps us apply math to real-life situations. Plus, we'll touch on algebraic expressions and rational equations, which are key building blocks for more advanced math concepts.

Solving Equations with Fractions

Verification of fraction equations

• Substitute the variable with the given fraction to check if it satisfies the equation
  • Perform necessary arithmetic operations to simplify the equation ($\frac{2}{3}x + \frac{1}{4} = \frac{7}{12}$, substitute $x$ with $\frac{1}{2}$)
  • Evaluate the resulting equation to determine if it is true or false
    • True: the fraction satisfies the equation ($\frac{2}{3} \cdot \frac{1}{2} + \frac{1}{4} = \frac{1}{3} + \frac{1}{4} = \frac{7}{12}$, true)
    • False: the fraction does not satisfy the equation ($\frac{2}{3} \cdot \frac{3}{4} + \frac{1}{4} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \neq \frac{7}{12}$, false)

Solving equations with fractions

• Find the least common denominator (LCD) of all fractions in the equation ($\frac{1}{2}x + \frac{1}{3} = \frac{1}{4}$, LCD is 12)
  • Multiply both sides of the equation by the LCD to eliminate fractions ($12 \cdot (\frac{1}{2}x + \frac{1}{3}) = 12 \cdot \frac{1}{4}$)
• Perform addition, subtraction, or division to isolate the variable on one side
  • Combine like terms and simplify the equation ($6x + 4 = 3$)
  • Subtract 4 from both sides: $6x = -1$
• Solve the equation for the variable by dividing both sides by the coefficient ($x = -\frac{1}{6}$)
  • This process is known as variable isolation

Multiplication property for fraction equations

• Multiply both sides of the equation by the reciprocal of the fraction coefficient ($\frac{2}{3}x = \frac{1}{4}$, multiply by $\frac{3}{2}$)
  • The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$ ($\frac{3}{2}$ is the reciprocal of $\frac{2}{3}$)
  • This step eliminates the fraction coefficient of the variable ($\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot \frac{1}{4}$)
• Simplify the equation by performing the necessary arithmetic operations ($x = \frac{3}{8}$)
• Solve the equation for the variable (already solved in the previous step)

Word problems to fraction equations

• Identify the unknown quantity and assign a variable to represent it (let $x$ be the number of apples)
• Translate the word problem into an equation using the given information
  • Express relationships using mathematical operations (half of the apples plus 3 equals 7)
  • Use fractions to represent parts of a whole or ratios ($\frac{1}{2}x + 3 = 7$)
• Solve the equation using the appropriate method (subtract 3 from both sides: $\frac{1}{2}x = 4$, multiply by 2: $x = 8$)
  • Follow the steps outlined in the previous objectives to solve the equation
• Interpret the solution in the context of the word problem
  • Check if the solution makes sense and satisfies the given conditions (8 apples, when halved and added to 3, equals 7)

Working with Algebraic Expressions and Rational Equations

• Algebraic expressions involve variables, numbers, and mathematical operations (e.g., $2x + 3$, $\frac{x}{2} - 1$)
• Rational equations are equations that contain fractions with variables in the denominator
• When solving rational equations, find common denominators to simplify the equation
• Apply the same problem-solving techniques used for other fraction equations