Solving Equations with Fractions
Solving equations with fractions is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students eliminate fractions by multiplying both sides by the LCD, then solve the resulting equation. This technique simplifies complex fractional equations into manageable integer equations.
Key Concepts
Property To solve an equation with fractions, we multiply each side of the equation by the denominator of the fraction. This will clear the fraction and give us an equivalent equation without fractions. If the equation contains more than one fraction, we can clear all the denominators at once by multiplying both sides by the LCD of the fractions. We must multiply each term of an equation by the LCD, whether or not the term is a fraction.
Examples To solve $\frac{50}{10 x} = 5$, we multiply both sides by $10 x$ to get $50 = 5(10 x)$. Distributing gives $50 = 50 5x$, which simplifies to $5x=0$, so $x=0$.
Solve $\frac{5}{x} 2 = \frac{7}{2x + 1}$. The LCD is $x(2x+1)$. Multiplying each term gives $x(2x+1)(\frac{5}{x}) x(2x+1)(2) = x(2x+1)(\frac{7}{2x+1})$. This simplifies to $5(2x+1) 2x(2x+1) = 7x$, which becomes a quadratic equation to solve.
Common Questions
How do you solve an equation that has fractions?
Find the least common denominator (LCD) of all fractions, multiply every term by the LCD to clear all fractions, then solve the resulting integer equation.
What is the LCD and how do you find it?
The LCD (least common denominator) is the smallest number divisible by all denominators in the equation. Find it by listing multiples or using prime factorization.
What should you check after solving an equation with fractions?
Check for extraneous solutions by substituting your answer back into the original equation, especially when the variable appears in a denominator.
How do you solve (x/3) + (x/4) = 7?
The LCD is 12. Multiply through: 4x + 3x = 84, so 7x = 84, giving x = 12.