Solving Equations: -24 = (4/7)x + Solution Verification

by ADMIN 56 views

Let's dive into solving the equation $-24 = \frac{4}{7} x$, and then we'll double-check our answer to make sure we got it right. Solving equations is a fundamental skill in algebra, and mastering it will help you tackle more complex problems down the road. So, buckle up, and let's get started!

Solving the Equation

Our main goal here is to isolate x on one side of the equation. Right now, x is being multiplied by $\frac{4}{7}$. To undo this multiplication, we need to perform the inverse operation, which is division. However, instead of dividing by a fraction (which can be a bit tricky), we'll multiply by the reciprocal of the fraction. The reciprocal of $\frac{4}{7}$ is $\frac{7}{4}$.

So, we'll multiply both sides of the equation by $\frac{7}{4}$:

74×(−24)=74×47x\frac{7}{4} \times (-24) = \frac{7}{4} \times \frac{4}{7} x

On the right side, the $\frac{7}{4}$ and $\frac{4}{7}$ cancel each other out, leaving us with just x:

74×(−24)=x\frac{7}{4} \times (-24) = x

Now, let's simplify the left side. We have $\frac{7}{4} \times (-24)$. We can think of -24 as $\frac{-24}{1}$, so we have:

74×−241=x\frac{7}{4} \times \frac{-24}{1} = x

Multiply the numerators and the denominators:

7×−244×1=x\frac{7 \times -24}{4 \times 1} = x

−1684=x\frac{-168}{4} = x

Now, divide -168 by 4:

−42=x-42 = x

So, our solution is x = -42.

Key Points Recap:

  • To isolate x, we multiplied both sides of the equation by the reciprocal of $\frac{4}{7}$, which is $\frac{7}{4}$.
  • We simplified the equation step-by-step to arrive at x = -42.

Checking the Solution

Now that we've found a solution, it's super important to check if it's correct. To do this, we'll substitute x = -42 back into the original equation:

−24=47x-24 = \frac{4}{7} x

Replace x with -42:

−24=47×(−42)-24 = \frac{4}{7} \times (-42)

We can think of -42 as $\frac{-42}{1}$, so we have:

−24=47×−421-24 = \frac{4}{7} \times \frac{-42}{1}

Multiply the numerators and the denominators:

−24=4×−427×1-24 = \frac{4 \times -42}{7 \times 1}

−24=−1687-24 = \frac{-168}{7}

Now, divide -168 by 7:

−24=−24-24 = -24

Since the left side of the equation equals the right side, our solution x = -42 is correct!

Verification Recap:

  • We substituted x = -42 back into the original equation.
  • We simplified both sides of the equation.
  • We confirmed that both sides are equal, verifying our solution.

Why is Checking Solutions Important?

Checking your solutions is a crucial step in solving equations. It helps you catch any mistakes you might have made during the solving process. It's like having a safety net – it ensures that you're not just getting an answer, but the correct answer. This is especially important in exams or when dealing with real-world problems where accuracy is key. By verifying your solutions, you build confidence in your problem-solving abilities and avoid potential errors.

Additional Tips for Solving Equations

  • Simplify: Before you start solving, simplify both sides of the equation as much as possible. This might involve combining like terms or distributing values.
  • Isolate the Variable: The goal is always to get the variable by itself on one side of the equation. Use inverse operations (addition/subtraction, multiplication/division) to undo any operations that are being performed on the variable.
  • Check Your Work: Always double-check your solution by plugging it back into the original equation.
  • Stay Organized: Keep your work neat and organized. This will help you avoid mistakes and make it easier to spot any errors if they occur.
  • Practice Regularly: The more you practice solving equations, the better you'll become at it. Start with simple equations and gradually work your way up to more complex ones.

Common Mistakes to Avoid

  • Forgetting to Distribute: When you have a number or variable outside of parentheses, make sure to distribute it to all terms inside the parentheses.
  • Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine 3x and 5x, but you cannot combine 3x and 5x². Double check your combinations to prevent this from happening!
  • Incorrectly Applying Inverse Operations: Make sure you're using the correct inverse operation to undo an operation. For example, to undo addition, you need to subtract; to undo multiplication, you need to divide.
  • Not Checking Your Solution: As we've emphasized, always check your solution to make sure it's correct. This is the best way to catch any mistakes you might have made.
  • Sign Errors: Pay close attention to signs (positive and negative). A simple sign error can throw off your entire solution. Double-check your calculations to avoid this. Always remember, a positive times a negative is a negative. Also, a negative times a negative is a positive, so keep a close eye on those pesky negative signs!

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

  1. Solve for y: $3y + 5 = 14$
  2. Solve for a: $-2a - 7 = 3$
  3. Solve for z: $\frac{z}{5} + 2 = 8$

Remember to check your solutions after you solve each equation!

Conclusion

So, there you have it! We've successfully solved the equation $-24 = \frac{4}{7} x$ and verified that our solution, x = -42, is correct. Remember to always check your work and practice regularly to improve your equation-solving skills. Keep practicing, and you'll become a pro at solving equations in no time! Keep an eye out for more guides coming your way!