Solving For Z (z-7)/(z-4) = (z-2)/(z-6) + 1 A Step-by-Step Guide
Hey everyone! Today, we're diving into a fun math problem where we need to solve for the variable z in the equation:
(z-7)/(z-4) = (z-2)/(z-6) + 1
This looks a bit tricky at first, but don't worry, we'll break it down step by step and make it super easy to understand. So, grab your pencils, paper, and let's get started!
Step 1: Get Rid of Those Fractions!
The first thing we want to do is eliminate the fractions. Fractions can sometimes make equations look more complicated than they actually are. To do this, we'll find the least common denominator (LCD) of the fractions in the equation. In our case, the denominators are (z-4) and (z-6). Since these are different expressions, the LCD will simply be their product: (z-4)(z-6).
Now, we'll multiply both sides of the equation by this LCD. Remember, what we do to one side of the equation, we must do to the other side to keep it balanced. This gives us:
(z-4)(z-6) * [(z-7)/(z-4)] = (z-4)(z-6) * [(z-2)/(z-6) + 1]
On the left side, the (z-4) terms cancel out, leaving us with:
(z-6)(z-7)
On the right side, we need to distribute the (z-4)(z-6) term to both parts of the expression. When we distribute to the first term, (z-2)/(z-6), the (z-6) terms cancel out, and when we distribute to the second term, which is just 1, we simply multiply (z-4)(z-6) by 1. This gives us:
(z-4)(z-6) * [(z-2)/(z-6)] + (z-4)(z-6) * 1
Simplifying, we get:
(z-4)(z-2) + (z-4)(z-6)
So, our equation now looks like this:
(z-6)(z-7) = (z-4)(z-2) + (z-4)(z-6)
Step 2: Expand Those Parentheses!
Next up, we need to expand the parentheses on both sides of the equation. This means multiplying out the terms inside the parentheses. Let's start with the left side, (z-6)(z-7). Using the distributive property (also known as FOIL – First, Outer, Inner, Last), we get:
z * z - 7 * z - 6 * z + 6 * 7
Simplifying, this becomes:
z^2 - 13z + 42
Now, let's expand the first part of the right side, (z-4)(z-2). Again, using the distributive property, we get:
z * z - 2 * z - 4 * z + 4 * 2
Which simplifies to:
z^2 - 6z + 8
Finally, let's expand the last part of the right side, (z-4)(z-6). Using the distributive property one more time, we get:
z * z - 6 * z - 4 * z + 4 * 6
Which simplifies to:
z^2 - 10z + 24
So, our equation now looks like this:
z^2 - 13z + 42 = (z^2 - 6z + 8) + (z^2 - 10z + 24)
Step 3: Simplify and Combine Like Terms
Now it's time to simplify our equation by combining like terms on the right side. We have two z^2 terms, a -6z and a -10z term, and the constants 8 and 24. Combining these, we get:
z^2 + z^2 = 2z^2
-6z - 10z = -16z
8 + 24 = 32
So, the right side of the equation becomes:
2z^2 - 16z + 32
Our equation now looks like this:
z^2 - 13z + 42 = 2z^2 - 16z + 32
Step 4: Move Everything to One Side
To solve for z, we need to get all the terms on one side of the equation, leaving zero on the other side. Let's move everything to the right side. We'll subtract z^2 from both sides, add 13z to both sides, and subtract 42 from both sides. This gives us:
0 = 2z^2 - z^2 - 16z + 13z + 32 - 42
Simplifying, we get:
0 = z^2 - 3z - 10
Step 5: Solve the Quadratic Equation
We now have a quadratic equation in the form az^2 + bz + c = 0, where a = 1, b = -3, and c = -10. There are a few ways to solve quadratic equations, but the most common methods are factoring, using the quadratic formula, or completing the square. In this case, factoring is the easiest approach.
We need to find two numbers that multiply to -10 and add up to -3. These numbers are -5 and 2. So, we can factor the quadratic equation as follows:
0 = (z - 5)(z + 2)
Now, to find the solutions for z, we set each factor equal to zero:
z - 5 = 0 or z + 2 = 0
Solving for z in each equation, we get:
z = 5 or z = -2
Step 6: Check for Extraneous Solutions
It's super important to check our solutions in the original equation to make sure they are valid. Sometimes, solutions we find might not actually work because they could make the denominator of a fraction equal to zero, which is undefined. These are called extraneous solutions.
Let's check z = 5 in the original equation:
(5 - 7)/(5 - 4) = (5 - 2)/(5 - 6) + 1
(-2)/1 = 3/(-1) + 1
-2 = -3 + 1
-2 = -2
So, z = 5 is a valid solution.
Now, let's check z = -2 in the original equation:
(-2 - 7)/(-2 - 4) = (-2 - 2)/(-2 - 6) + 1
(-9)/(-6) = (-4)/(-8) + 1
3/2 = 1/2 + 1
3/2 = 3/2
So, z = -2 is also a valid solution.
Final Answer:
We found two solutions for z that both check out in the original equation. Therefore, the solutions are:
z = 5, -2
And that's it! We've successfully solved for z in the equation. Great job, guys! Remember, the key to solving these types of problems is to take it step by step, eliminate the fractions, expand the parentheses, simplify, and check your solutions. Keep practicing, and you'll become a pro at solving equations in no time!
Are you struggling with equations that have variables in the denominator? Do you find yourself getting lost in the steps and making mistakes? Don't worry; you're not alone! Solving for z in equations like the one we tackled earlier can seem daunting, but with a systematic approach and a bit of practice, you can conquer even the most complex equations. In this comprehensive guide, we'll delve deeper into the techniques and strategies needed to solve for z effectively. So, let's sharpen our pencils, fire up our brains, and get ready to become equation-solving masters!
Understanding the Fundamentals
Before we dive into advanced techniques, it's crucial to solidify our understanding of the fundamental principles involved in solving equations. At its core, solving an equation means finding the value (or values) of the variable that makes the equation true. Think of an equation as a balanced scale, with the left side equaling the right side. Our goal is to isolate the variable on one side of the equation while maintaining this balance.
Key Principles to Remember:
- The Golden Rule of Algebra: What you do to one side of the equation, you must do to the other side. This is the cornerstone of equation-solving and ensures that the equation remains balanced.
- Inverse Operations: To isolate the variable, we use inverse operations. Addition and subtraction are inverse operations, as are multiplication and division. For example, to undo addition, we subtract; to undo multiplication, we divide.
- Order of Operations: When simplifying expressions, remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Identifying the Challenge: Fractions and Rational Expressions
The equation we solved earlier, (z-7)/(z-4) = (z-2)/(z-6) + 1, presents a common challenge: fractions with variables in the denominator. These are known as rational expressions, and they require a specific approach to solve effectively. The main hurdle is eliminating the fractions to simplify the equation and make it easier to manipulate.
Strategies for Solving Equations with Fractions
1. Finding the Least Common Denominator (LCD)
The first and most crucial step in solving equations with fractions is to find the least common denominator (LCD). The LCD is the smallest multiple that all the denominators in the equation divide into evenly. This allows us to multiply both sides of the equation by the LCD, effectively eliminating the fractions.
How to Find the LCD:
- Factor each denominator completely.
- Identify all the unique factors present in the denominators.
- For each unique factor, take the highest power that appears in any of the denominators.
- Multiply these highest powers together to get the LCD.
In our example, the denominators are (z-4) and (z-6). Since these are linear expressions with no common factors, the LCD is simply their product: (z-4)(z-6).
2. Multiplying by the LCD
Once we've found the LCD, the next step is to multiply both sides of the equation by it. This is where the magic happens! Multiplying by the LCD will cancel out the denominators, leaving us with a simpler equation to work with.
Distribute Carefully:
When multiplying by the LCD, remember to distribute it to each term on both sides of the equation. This is a critical step to ensure that the equation remains balanced.
In our example, we multiplied both sides of the equation by (z-4)(z-6), which resulted in the cancellation of the denominators and a significant simplification of the equation.
3. Expanding and Simplifying
After eliminating the fractions, the next step is to expand any parentheses and simplify the resulting expressions. This often involves using the distributive property (or FOIL method) to multiply out terms and then combining like terms to reduce the equation to its simplest form.
Pay Attention to Signs:
Be extra careful with negative signs when expanding and simplifying. A small mistake in sign can lead to an incorrect solution.
In our example, we expanded the products (z-6)(z-7), (z-4)(z-2), and (z-4)(z-6) using the distributive property and then combined like terms to simplify the equation.
4. Solving the Equation
Once the equation is simplified, we can use standard algebraic techniques to solve for z. The specific method we use will depend on the type of equation we have. In our example, we ended up with a quadratic equation, which we solved by factoring. However, other types of equations may require different approaches.
Common Equation Types and Solution Methods:
- Linear Equations: Use inverse operations to isolate the variable.
- Quadratic Equations: Factor, use the quadratic formula, or complete the square.
- Rational Equations: Clear fractions, simplify, and solve the resulting equation.
- Radical Equations: Isolate the radical, square (or cube, etc.) both sides, and solve the resulting equation.
5. Checking for Extraneous Solutions
This is a crucial step that is often overlooked, but it can save you from getting the wrong answer. When solving equations with fractions or radicals, it's essential to check our solutions in the original equation to make sure they are valid. Sometimes, solutions we find may not actually work because they could make a denominator zero or result in taking the square root of a negative number. These are called extraneous solutions.
Why Extraneous Solutions Occur:
Extraneous solutions arise because certain algebraic manipulations, such as squaring both sides of an equation or multiplying by an expression that could be zero, can introduce solutions that don't satisfy the original equation.
In our example, we checked both z = 5 and z = -2 in the original equation and confirmed that they were both valid solutions.
Advanced Tips and Techniques
1. Recognizing Patterns and Shortcuts
As you solve more equations, you'll start to recognize patterns and shortcuts that can save you time and effort. For example, if you see a common factor in multiple terms, you can factor it out to simplify the equation. Or, if you notice that the equation is in a specific form (such as a difference of squares), you can use a shortcut formula to factor it quickly.
2. Using Substitution
In some cases, you may encounter equations that are complex but have a repeating pattern. In these situations, using substitution can be a helpful technique. We can substitute a single variable for the repeating expression, solve for that variable, and then substitute back to find the value of z.
3. Graphing the Equation
If you're struggling to solve an equation algebraically, graphing it can sometimes provide valuable insights. The solutions to the equation will correspond to the points where the graph intersects the x-axis. Graphing can also help you visualize the equation and identify any potential extraneous solutions.
4. Practice, Practice, Practice!
The most important tip for mastering equation-solving is to practice consistently. The more equations you solve, the more comfortable you'll become with the techniques and strategies involved. Start with simpler equations and gradually work your way up to more challenging ones. Don't be afraid to make mistakes – they are a natural part of the learning process. Just be sure to learn from your mistakes and keep practicing!
Conclusion
Solving for z in equations, especially those with fractions and rational expressions, requires a systematic approach and a solid understanding of algebraic principles. By following the steps outlined in this guide – finding the LCD, multiplying by the LCD, expanding and simplifying, solving the equation, and checking for extraneous solutions – you can confidently tackle even the most challenging equations. Remember to practice regularly, look for patterns and shortcuts, and don't be afraid to ask for help when you need it. With dedication and perseverance, you'll become an equation-solving expert in no time!
Hey everyone! We've covered a lot about solving for z in various equations. Now, let's talk about some common pitfalls that students often encounter. Recognizing these mistakes beforehand can save you from headaches and help you ace those math problems! So, let's dive into the most frequent errors and how to steer clear of them.
1. Forgetting to Distribute Properly
One of the most common mistakes is not distributing properly when multiplying expressions. Remember, the distributive property states that a(b + c) = ab + ac. This means you need to multiply the term outside the parentheses by every term inside the parentheses.
Example of the Mistake:
Let's say you have the expression 2(z + 3). A common mistake is to multiply only the z by 2, resulting in 2z + 3. This is incorrect! You need to multiply both the z and the 3 by 2.
The Correct Way:
2(z + 3) = 2 * z + 2 * 3 = 2z + 6
How to Avoid This Mistake:
- Slow Down: Take your time and make sure you're multiplying each term inside the parentheses by the term outside.
- Write it Out: If it helps, write out each multiplication step explicitly (e.g., 2 * z + 2 * 3).
- Double-Check: After distributing, quickly check to make sure you've multiplied each term correctly.
2. Incorrectly Combining Like Terms
Combining like terms is a crucial step in simplifying equations, but it's also a place where mistakes can easily happen. Like terms are terms that have the same variable raised to the same power (e.g., 3z and -5z, or 2z^2 and 7z^2). You can only combine like terms by adding or subtracting their coefficients.
Example of the Mistake:
Suppose you have the expression 3z + 2 + 5z - 1. A common mistake is to incorrectly combine terms, such as adding 3z and 2 or forgetting to account for negative signs.
The Correct Way:
Identify the like terms: 3z and 5z are like terms, and 2 and -1 are like terms. Combine them: 3z + 5z = 8z and 2 - 1 = 1. So, the simplified expression is 8z + 1.
How to Avoid This Mistake:
- Identify Like Terms: Before combining, clearly identify the terms that are alike.
- Pay Attention to Signs: Be careful with positive and negative signs. Remember, subtraction is the same as adding a negative number.
- Rearrange if Necessary: If it helps, rearrange the expression so that like terms are next to each other (e.g., 3z + 5z + 2 - 1).
3. Sign Errors
Sign errors are a classic mistake in algebra, and they can completely throw off your answer. These errors usually occur when dealing with negative numbers or when distributing a negative sign.
Example of the Mistake:
Let's say you need to simplify the expression 5 - (2z - 3). A common mistake is to forget to distribute the negative sign to both terms inside the parentheses, resulting in 5 - 2z - 3. This is incorrect!
The Correct Way:
Distribute the negative sign to both terms inside the parentheses: 5 - (2z - 3) = 5 - 2z + 3. Then, combine like terms: 5 + 3 = 8. So, the simplified expression is 8 - 2z.
How to Avoid This Mistake:
- Treat Subtraction as Adding a Negative: Rewrite subtraction as adding a negative (e.g., a - b = a + (-b)).
- Be Careful When Distributing Negatives: When distributing a negative sign, make sure you change the sign of every term inside the parentheses.
- Double-Check: After distributing, double-check that you've handled the signs correctly.
4. Forgetting to Perform the Same Operation on Both Sides
The Golden Rule of Algebra states that what you do to one side of the equation, you must do to the other side. This is essential for maintaining balance and solving equations correctly. Forgetting this rule can lead to incorrect solutions.
Example of the Mistake:
Suppose you have the equation 2z + 5 = 11. To solve for z, you need to subtract 5 from both sides. A common mistake is to subtract 5 only from the left side, resulting in 2z = 11, which leads to an incorrect solution.
The Correct Way:
Subtract 5 from both sides: 2z + 5 - 5 = 11 - 5. This simplifies to 2z = 6. Then, divide both sides by 2: 2z / 2 = 6 / 2, which gives you z = 3.
How to Avoid This Mistake:
- Write it Down: When performing an operation on one side of the equation, immediately write down the same operation on the other side.
- Check Your Work: After each step, double-check that you've performed the same operation on both sides.
- Visualize the Balance: Think of the equation as a balanced scale. If you add or remove something from one side, you need to do the same on the other side to maintain balance.
5. Not Checking for Extraneous Solutions
As we discussed earlier, when solving equations with fractions or radicals, it's crucial to check for extraneous solutions. These are solutions that you obtain algebraically but don't actually satisfy the original equation.
Example of the Mistake:
We saw an example of this in our original problem. Failing to check the solutions would mean not knowing if the values we obtained for z were correct.
The Correct Way:
Always substitute your solutions back into the original equation to verify that they work. If a solution makes the denominator of a fraction zero or leads to taking the square root of a negative number, it's an extraneous solution and should be discarded.
How to Avoid This Mistake:
- Make it a Habit: Always check your solutions in the original equation, especially when dealing with fractions or radicals.
- Be Thorough: Substitute each solution individually and simplify both sides of the equation to see if they are equal.
- Discard Extraneous Solutions: If a solution doesn't work in the original equation, clearly indicate that it's an extraneous solution.
Conclusion
By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence when solving equations for z. Remember to take your time, double-check your work, and practice consistently. With these tips in mind, you'll be solving equations like a pro in no time! Happy calculating!