Solve Equations: 8x-24=y-3 & 9x+9=2y+2

by Mei Lin 39 views

Hey everyone! Let's dive into solving this system of equations. It looks a bit intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand. We're going to tackle the equations 8x - 24 = y - 3 and 9x + 9 = 2y + 2 to find the values of x and y that satisfy both. Get ready to sharpen those math skills and unlock the solution! Our final answer must match one of the options provided: a) (7; 35), b) (6; 10), c) (2; 7), d) (7; 5), or e) N.A. (Not Applicable).

Step-by-Step Solution

Let's start by simplifying the given equations. This will make them easier to work with and help us see the relationships between x and y more clearly. Remember, the key to solving any math problem is to take it one step at a time!

1. Simplify the First Equation: 8x - 24 = y - 3

Our first equation is 8x - 24 = y - 3. To simplify, we'll isolate y on one side. We can do this by adding 3 to both sides of the equation:

8x - 24 + 3 = y - 3 + 3

This simplifies to:

8x - 21 = y

So, we now have our first equation in a cleaner form: y = 8x - 21. This is a crucial step as it gives us a direct relationship between y and x.

2. Simplify the Second Equation: 9x + 9 = 2y + 2

Next up is the second equation: 9x + 9 = 2y + 2. Similar to the first equation, let's isolate y. First, subtract 2 from both sides:

9x + 9 - 2 = 2y + 2 - 2

This gives us:

9x + 7 = 2y

Now, divide both sides by 2 to completely isolate y:

(9x + 7) / 2 = y

So, our second equation in simplified form is: y = (9x + 7) / 2. Now we have two equations where y is expressed in terms of x, which sets us up perfectly for the next step.

3. Solve for x by Equating the Expressions for y

Now that we have both equations in the form y = something, we can set those expressions equal to each other. This is because, at the solution point, the y values must be the same. So, we have:

8x - 21 = (9x + 7) / 2

To get rid of the fraction, multiply both sides of the equation by 2:

2 * (8x - 21) = 2 * ((9x + 7) / 2)

This simplifies to:

16x - 42 = 9x + 7

Now, let's isolate x. Subtract 9x from both sides:

16x - 9x - 42 = 9x - 9x + 7

Which gives us:

7x - 42 = 7

Next, add 42 to both sides:

7x - 42 + 42 = 7 + 42

This simplifies to:

7x = 49

Finally, divide both sides by 7 to solve for x:

x = 49 / 7

So, we find that x = 7. Great job! We've found the value of x. Now, let's use this to find y.

4. Solve for y by Substituting the Value of x

We know that x = 7. We can plug this value into either of our simplified equations to find y. Let's use the first one, y = 8x - 21, as it looks a bit simpler:

y = 8 * 7 - 21

y = 56 - 21

y = 35

So, y = 35. Fantastic! We've found both x and y.

5. Verify the Solution

Before we declare victory, it's always a good idea to check our solution. Let's plug x = 7 and y = 35 into both original equations to make sure they hold true.

Equation 1: 8x - 24 = y - 3

8 * 7 - 24 = 35 - 3

56 - 24 = 32

32 = 32 (This is true!)

Equation 2: 9x + 9 = 2y + 2

9 * 7 + 9 = 2 * 35 + 2

63 + 9 = 70 + 2

72 = 72 (This is also true!)

Our solution checks out in both equations. We can be confident that we've found the correct values for x and y.

Final Answer

We found that x = 7 and y = 35. Therefore, the solution to the system of equations is the ordered pair (7, 35). Looking back at our options, this corresponds to:

a) (7; 35)

So, the correct answer is (7; 35). You did it! Solving systems of equations might seem tough at first, but with practice and a step-by-step approach, you can conquer any math challenge. Keep up the great work!

Why This Problem Matters

You might be wondering, “Why do we even need to solve systems of equations?” Well, these skills are crucial in many real-world applications. From engineering and physics to economics and computer science, systems of equations help us model and solve complex problems. For example, engineers might use them to design structures, economists to predict market trends, and computer scientists to develop algorithms. Mastering these concepts opens up a world of possibilities and strengthens your problem-solving abilities in all areas of life.

Different Methods for Solving Systems of Equations

While we used the substitution method here, there are other techniques you can use to solve systems of equations. Let's briefly touch on a couple of them:

1. Elimination Method

The elimination method involves manipulating the equations so that when you add or subtract them, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve. Then, you substitute the value back into one of the original equations to find the other variable. The elimination method can be particularly useful when the coefficients of one variable are multiples of each other.

2. Graphing Method

Each equation in a system represents a line. The solution to the system is the point where the lines intersect. The graphing method involves plotting the lines and visually identifying their intersection point. While this method can be helpful for visualizing the solution, it might not be the most accurate for non-integer solutions. However, it provides a great way to understand the concept of a solution graphically.

Tips for Tackling System of Equations Problems

Solving systems of equations can become second nature with practice. Here are a few tips to keep in mind:

  • Simplify first: Always simplify the equations as much as possible before attempting to solve them. This will make the calculations easier and reduce the chances of errors.
  • Choose the right method: Consider the structure of the equations and choose the method that seems most efficient. Sometimes substitution is easier, while other times elimination might be the way to go.
  • Check your solution: As we did in this problem, always verify your solution by plugging the values back into the original equations. This ensures that you haven't made any mistakes along the way.
  • Practice, practice, practice: The more you practice, the more comfortable you'll become with different types of problems and techniques.

Keep these tips in mind, and you'll be solving systems of equations like a pro in no time!

Conclusion

So, there you have it! We successfully solved the system of equations 8x - 24 = y - 3 and 9x + 9 = 2y + 2, finding the solution (7, 35). Remember, the key is to break down the problem into manageable steps, simplify the equations, and choose the appropriate method. Keep practicing, and you'll become a master of solving systems of equations. And remember, math can be fun when you approach it with the right mindset and a bit of perseverance. Keep exploring, keep learning, and keep solving!