Solve For X: Fraction Equation Tutorial

by Henrik Larsen 40 views

Hey guys! Let's tackle this equation and find the value of x. We've got a fraction problem here, but don't worry, we'll break it down step by step. This is a classic algebra problem that involves isolating the variable x by performing operations on both sides of the equation. So, grab your pencils, and let's get started!

The Problem

Our mission, should we choose to accept it (and we do!), is to solve for x in the following equation:

34=12−79x\frac{3}{4} = \frac{1}{2} - \frac{7}{9}x

We need to isolate x on one side of the equation. To do that, we'll use inverse operations. Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced.

Step 1: Isolate the Term with x

The first thing we want to do is get the term with x (which is −79x-\frac{7}{9}x) by itself on one side of the equation. Currently, we have 12\frac{1}{2} being added to it. To get rid of this 12\frac{1}{2}, we'll subtract 12\frac{1}{2} from both sides of the equation. This is the addition property of equality in action – we're maintaining the balance by doing the same thing on both sides.

34−12=12−79x−12\frac{3}{4} - \frac{1}{2} = \frac{1}{2} - \frac{7}{9}x - \frac{1}{2}

This simplifies to:

34−12=−79x\frac{3}{4} - \frac{1}{2} = -\frac{7}{9}x

Now, before we can proceed, we need to subtract the fractions on the left side. Remember, we can only add or subtract fractions if they have a common denominator. So, let's find the least common multiple (LCM) of 4 and 2. The LCM of 4 and 2 is 4. We'll rewrite 12\frac{1}{2} with a denominator of 4.

12=1×22×2=24\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}

Now we can substitute this back into our equation:

34−24=−79x\frac{3}{4} - \frac{2}{4} = -\frac{7}{9}x

Subtracting the fractions on the left side, we get:

14=−79x\frac{1}{4} = -\frac{7}{9}x

Great! We've isolated the term with x. Now, let's move on to the next step.

Step 2: Get Rid of the Fraction Coefficient

Okay, so now we have 14=−79x\frac{1}{4} = -\frac{7}{9}x. Our goal is to get x all by itself. Notice that x is being multiplied by −79-\frac{7}{9}. To undo this multiplication, we'll multiply both sides of the equation by the reciprocal of −79-\frac{7}{9}, which is −97-\frac{9}{7}. Multiplying by the reciprocal is the same as dividing, and it's a neat trick to get rid of fractional coefficients!

(−97)×14=(−97)×(−79x)(-\frac{9}{7}) \times \frac{1}{4} = (-\frac{9}{7}) \times (-\frac{7}{9}x)

On the right side, the −79-\frac{7}{9} and −97-\frac{9}{7} cancel each other out, leaving us with just x. On the left side, we multiply the fractions:

−9×17×4=x-\frac{9 \times 1}{7 \times 4} = x

This simplifies to:

−928=x-\frac{9}{28} = x

Step 3: The Solution

And there we have it! We've solved for x. Our answer is:

x=−928x = -\frac{9}{28}

This is a proper fraction since the absolute value of the numerator (9) is less than the absolute value of the denominator (28). It's also in simplest terms because 9 and 28 have no common factors other than 1. So, we're all done!

Key Concepts Used

Let's recap the main concepts we used to solve this equation:

  • Addition Property of Equality: Adding or subtracting the same value from both sides of an equation maintains the equality.
  • Least Common Multiple (LCM): Finding the LCM allows us to add or subtract fractions with different denominators.
  • Reciprocal: The reciprocal of a fraction ab\frac{a}{b} is ba\frac{b}{a}. Multiplying a fraction by its reciprocal results in 1.
  • Multiplication Property of Equality: Multiplying or dividing both sides of an equation by the same non-zero value maintains the equality.

Practice Makes Perfect

Solving equations with fractions might seem tricky at first, but with practice, you'll become a pro! Try solving similar equations on your own. The more you practice, the more comfortable you'll become with the steps involved. Remember to always check your work by plugging your solution back into the original equation to make sure it holds true. Keep up the great work, guys! You've got this!

Let's Talk Strategy

When solving equations like this, it's super helpful to have a strategy in mind. Think of it like a game plan. Here’s the general strategy we used, and you can use it for other similar problems:

  1. Simplify Both Sides: If there’s any simplifying you can do on either side of the equation first (like combining like terms), do that. It'll make the problem cleaner and easier to work with.

  2. Isolate the Variable Term: Our main goal is to get the term with the variable (in this case, x) alone on one side. To do this, we use inverse operations. We added 12\frac{1}{2}'s opposite to both sides to get the x term by itself.

  3. Get Rid of Coefficients: Once the variable term is isolated, we need to deal with any coefficients (the number multiplied by the variable). If it’s a fraction, like we had here, multiply both sides by the reciprocal. If it’s a whole number, divide both sides by that number.

  4. Check Your Answer: Always, always, always check your answer! Plug it back into the original equation to make sure it works. This is a great way to catch any mistakes you might have made.

Real-World Connections

Now, you might be thinking,