Solving Algebraic Equations with Fractions - Step by Step

Welcome to solving equations with fractions! Today we'll learn the crucial first step: multiplying both sides by the LCD.Let's start with this equation: x over 3 plus 2 over 5 equals 4.The LCD, or Least Common Denominator, is the smallest number that's divisible by all denominators in our equation.In our equation, we have denominators of 3 and 5. Since these are both prime numbers, their LCD will be their product.Now that we have our LCD of 15, we multiply every term on both sides of the equation by 15.Using the distributive property, we multiply 15 by each term separately.Let's break down each multiplication. 15 times x over 3 simplifies to 5x, since 15 divided by 3 is 5. And 15 times 2 over 5 equals 6, since 15 divided by 5 is 3, then multiplied by 2.This gives us our final equation with no fractions: 5x plus 6 equals 60.We've successfully converted our fractional equation into a standard algebraic equation with integers.Starting with our integer equation, we'll solve step by step using standard algebraic techniques.First, we need to isolate the term with our variable x. We can do this by subtracting 6 from both sides.Remember, whatever operation we perform on one side of the equation, we must do to the other side to maintain equality.Now that we have isolated the term with x, we'll divide both sides by 5 to solve for x.When we divide fifty-four by five, we get our final answer: x equals fifty-four fifths.These three steps - isolating the variable term, removing other terms, and dividing by the coefficient - will help you solve any similar equation.Now that we have our solution, we'll need to verify it in the original equation.Now that we have our solution, let's verify it's correct by substituting it back into the original equation.First, we'll substitute our solution, fifty-four fifths, into the original equation.Next, we'll simplify the complex fraction by dividing fifty-four fifths by three.To add these fractions, we need a common denominator. The LCD is fifteen.Now we can add the numerators since the denominators are the same.Finally, we reduce sixty fifteenths to get our result.Since both sides are equal to four, our solution is verified correct!Here are some important tips to remember when verifying your solutions.And watch out for these common mistakes that students often make during verification.Remember, taking the time to verify your solution can help you catch any mistakes and ensure your answer is correct.