Solve: 3x + 10 = 22 & Find Possible Solutions

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Solve the equation 3x + 10 = 22 and find possible solutions

Hey guys! Let's break down this math problem step by step. We're going to figure out what real-world situation the equation 3x + 10 = 22 is describing. Then, we'll test the numbers 1, 2, 3, 4, and 5 to see which ones make the equation true. Grab your thinking caps, and let's dive in!

Understanding the Equation

So, what does 3x + 10 = 22 actually mean? Let's look at the parts:

  • 3x: This means "three times a number." The x is our mystery number, the one we're trying to find.
  • + 10: This means we're adding 10 to whatever 3x is.
  • = 22: This means the whole thing equals 22. So, after we multiply our mystery number by 3 and add 10, we should end up with 22.

With that in mind, let's evaluate the options you've given and see which one is more accurate with what was mentioned.

Option A: Ten more than three times a number is 22.

This one sounds right! If we take a number (x), multiply it by three (3x), and then add ten (+ 10), we get 22. This is exactly what the equation 3x + 10 = 22 says. So, option A matches perfectly.

Option B: If you subtract ten from three times a number, you get 22.

This isn't quite right. Subtracting 10 would look like 3x - 10 = 22. Our equation adds 10, so this option is a no-go.

Option C: If you add ten times a number, you get something....

This doesn't even fit the structure of our original equation, so it's incorrect.

Finding the Solution

Alright, we know what the equation means. Now, let's find out which of the numbers 1, 2, 3, 4, and 5 actually work in the equation. We'll do this by plugging each number in for x and seeing if the equation holds true.

  • If x = 1:
    • 3(1) + 10 = 3 + 10 = 13
    • 13 does not equal 22, so 1 is not a solution.
  • If x = 2:
    • 3(2) + 10 = 6 + 10 = 16
    • 16 does not equal 22, so 2 is not a solution.
  • If x = 3:
    • 3(3) + 10 = 9 + 10 = 19
    • 19 does not equal 22, so 3 is not a solution.
  • If x = 4:
    • 3(4) + 10 = 12 + 10 = 22
    • 22 equals 22! So, 4 is a solution!
  • If x = 5:
    • 3(5) + 10 = 15 + 10 = 25
    • 25 does not equal 22, so 5 is not a solution.

Wrapping Up

So, to recap, the equation 3x + 10 = 22 means "ten more than three times a number is 22." And out of the numbers 1, 2, 3, 4, and 5, only 4 is the solution that makes the equation true.

Hope this explanation helped you understand the problem better! Keep practicing, and you'll become a math whiz in no time! In summary:

  • The equation 3x + 10 = 22 represents the statement "Ten more than three times a number is 22."
  • The number 4 is the solution to the equation from the given options.

Extra Tips for Solving Equations

Solving equations is a fundamental skill in algebra, and mastering it can open doors to more advanced mathematical concepts. Here are some extra tips to help you tackle similar problems with confidence:

  1. Understand the Order of Operations: Always remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This ensures you perform operations in the correct sequence.

  2. Isolate the Variable: The main goal in solving an equation is to isolate the variable on one side of the equation. To do this, use inverse operations. For example, if you have x + 5 = 10, subtract 5 from both sides to get x = 5.

  3. Perform the Same Operation on Both Sides: Always maintain the balance of the equation by performing the same operation on both sides. Whether you're adding, subtracting, multiplying, or dividing, make sure to do it on both sides to keep the equation equal.

  4. Simplify Expressions: Before isolating the variable, simplify any expressions on either side of the equation. This might involve combining like terms or distributing a number across parentheses.

  5. Check Your Solution: After finding a potential solution, plug it back into the original equation to verify that it makes the equation true. This step helps you catch any mistakes you might have made along the way.

  6. Practice Regularly: Like any skill, solving equations requires practice. The more you practice, the more comfortable and confident you'll become. Try solving a variety of equations with different complexities to challenge yourself.

  7. Use Resources: Don't hesitate to use resources like textbooks, online tutorials, or math apps to help you understand concepts and practice problems. There are many excellent resources available to support your learning.

By following these tips and practicing regularly, you'll improve your equation-solving skills and build a strong foundation in algebra.

Real-World Applications of Equations

Equations aren't just abstract mathematical concepts; they have numerous real-world applications in various fields. Understanding how equations are used in real-life scenarios can make learning them more engaging and meaningful. Here are some examples:

  1. Finance: Equations are used to calculate interest rates, loan payments, and investment returns. For example, the formula for simple interest is I = PRT, where I is the interest, P is the principal, R is the rate, and T is the time. Equations help individuals and businesses make informed financial decisions.

  2. Physics: Physics relies heavily on equations to describe the relationships between physical quantities. For example, Newton's second law of motion is expressed as F = MA, where F is the force, M is the mass, and A is the acceleration. Equations help scientists understand and predict the behavior of the physical world.

  3. Engineering: Engineers use equations to design and analyze structures, circuits, and systems. For example, Ohm's law, V = IR, relates voltage (V), current (I), and resistance (R) in an electrical circuit. Equations are essential for ensuring the safety and efficiency of engineering projects.

  4. Chemistry: Chemistry uses equations to represent chemical reactions and calculate quantities of reactants and products. For example, a balanced chemical equation shows the ratio of reactants and products in a chemical reaction. Equations are fundamental to understanding and predicting chemical processes.

  5. Computer Science: Equations are used in algorithms, data analysis, and machine learning. For example, linear regression uses equations to model the relationship between variables and make predictions. Equations are crucial for developing efficient and accurate computational models.

  6. Everyday Life: Equations are also used in everyday life for tasks such as calculating gas mileage, determining the cost of items on sale, and converting units of measurement. For example, the equation distance = speed × time is used to calculate how far you can travel at a certain speed in a given amount of time.

By recognizing the real-world applications of equations, you can appreciate their importance and relevance beyond the classroom. This understanding can motivate you to learn and master equation-solving skills.

Additional Practice Problems

To further enhance your understanding and skills in solving equations, here are some additional practice problems. Try solving them on your own, and then check your answers to reinforce your learning.

  1. Solve for x: 5x - 8 = 12
  2. Solve for y: 2y + 7 = 15
  3. Solve for z: 3z - 4 = 8
  4. Solve for a: 4a + 9 = 21
  5. Solve for b: 6b - 5 = 19

Answers:

  1. x = 4
  2. y = 4
  3. z = 4
  4. a = 3
  5. b = 4

These practice problems cover a range of simple linear equations. Work through each problem step by step, and be sure to check your answers to ensure accuracy. Regular practice is key to mastering equation-solving skills.