Solving For Z: A Step-by-Step Guide

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Hey everyone! Today, we're diving into a common algebra problem: solving for a variable. In this case, we're tackling the equation 16z + 29z = pz - v and figuring out what 'z' equals. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you can follow along easily. This is a foundational skill in mathematics, and mastering it will help you in various fields like engineering, finance, and even computer science. Understanding how to manipulate equations and isolate variables opens doors to solving real-world problems, making it a valuable tool in your academic and professional journey.

Combining Like Terms

First things first, let's simplify both sides of the equation. On the left side, we have 16z + 29z. These are like terms because they both contain the variable 'z'. To combine them, we simply add their coefficients:

16z + 29z = (16 + 29)z = 45z

So, our equation now looks like this:

45z = pz - v

Combining like terms is a crucial step in simplifying equations. It allows us to reduce the complexity and make it easier to isolate the variable we're trying to solve for. By identifying and combining like terms, we can streamline the equation and work towards a solution more efficiently. This process is applicable to various algebraic expressions and is a fundamental skill in solving mathematical problems.

Isolating 'z' Terms

Our next goal is to get all the terms containing 'z' on one side of the equation. Currently, we have '45z' on the left and 'pz' on the right. To move 'pz' to the left side, we subtract it from both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the equality:

45z - pz = pz - v - pz

This simplifies to:

45z - pz = -v

Now, we have all the 'z' terms on the left side and the constant term '-v' on the right side. Isolating variables is a fundamental technique in algebra. By strategically adding or subtracting terms from both sides of the equation, we can group the terms containing the variable we want to solve for on one side and the constant terms on the other. This process allows us to simplify the equation and eventually isolate the variable, leading us closer to the solution.

Factoring out 'z'

Notice that 'z' is a common factor on the left side of the equation. We can factor it out:

z(45 - p) = -v

Factoring is the reverse process of distribution. It involves identifying a common factor in an expression and extracting it to simplify the expression. In this case, 'z' is a common factor in both terms on the left side of the equation. By factoring it out, we can rewrite the expression in a more compact form, which makes it easier to isolate 'z' in the next step. Factoring is a powerful tool in algebra that can simplify complex expressions and reveal underlying relationships between variables.

Solving for 'z'

Finally, to isolate 'z', we need to get rid of the (45 - p) term. We can do this by dividing both sides of the equation by (45 - p):

z(45 - p) / (45 - p) = -v / (45 - p)

This simplifies to:

z = -v / (45 - p)

So, we have solved for 'z'! The solution is z = -v / (45 - p). Division is the inverse operation of multiplication. By dividing both sides of the equation by the same non-zero expression, we can isolate the variable we're solving for. In this case, we divided both sides by (45 - p) to isolate 'z'. However, we need to be cautious when dividing by an expression containing a variable, as we need to ensure that the expression is not equal to zero. If (45 - p) were equal to zero, the division would be undefined, and the solution would not be valid.

Rewriting the Solution (Optional)

Sometimes, it's helpful to rewrite the solution in a slightly different form. We can multiply both the numerator and denominator by -1:

z = -v / (45 - p) = v / (p - 45)

This gives us an equivalent solution: z = v / (p - 45). Rewriting the solution can sometimes make it easier to interpret or compare with other solutions. In this case, by multiplying both the numerator and denominator by -1, we changed the sign of both the numerator and denominator, resulting in an equivalent expression. This can be useful if we want to avoid having a negative sign in the denominator or if we want to express the solution in a more concise form.

Important Note: When is the Solution Undefined?

It's crucial to remember that our solution is undefined if the denominator (45 - p) or (p - 45) equals zero. This would lead to division by zero, which is not allowed in mathematics.

So, the solution is undefined when:

p - 45 = 0

Solving for 'p', we get:

p = 45

Therefore, our solution z = v / (p - 45) is valid only when p ≠ 45. Understanding when a solution is undefined is essential in mathematics. It helps us identify potential limitations or restrictions on the values of variables. In this case, the solution is undefined when the denominator of the expression equals zero. By identifying these cases, we can ensure that our solutions are valid and meaningful.

Let's Recap

To solve for 'z' in the equation 16z + 29z = pz - v, we followed these steps:

  1. Combined like terms: 45z = pz - v
  2. Isolated 'z' terms: 45z - pz = -v
  3. Factored out 'z': z(45 - p) = -v
  4. Solved for 'z': z = -v / (45 - p) or z = v / (p - 45)
  5. Identified when the solution is undefined: p ≠ 45

Solving equations for a specific variable, like 'z' in our example, is a fundamental skill in mathematics with applications across various fields. By mastering these techniques, you can unlock the ability to solve complex problems and gain a deeper understanding of the relationships between variables. So, keep practicing and exploring different types of equations to strengthen your problem-solving skills.

Practice Makes Perfect

Solving for variables is a fundamental skill in algebra. The more you practice, the better you'll become. Try solving similar equations with different variables and constants to solidify your understanding. Don't be afraid to make mistakes – they are a part of the learning process. And remember, there are plenty of resources available online and in textbooks to help you along the way. So, keep practicing and exploring, and you'll become a pro at solving for variables in no time!

Keep up the great work, and you'll be solving complex equations in no time!