Solving Equations: Step-by-Step Guide With Examples

Hey guys! Let's dive into the world of equation-solving. It might seem daunting at first, but trust me, with a few simple steps, you'll be cracking these problems like a pro. We're going to tackle a range of equations, expressing our answers as either integers or fractions. So, grab your pencils, and let's get started!

Understanding the Basics of Solving Equations

Before we jump into specific problems, let's quickly review the core principles behind solving equations. The main goal is to isolate the variable (usually represented by letters like x, y, d, or r) on one side of the equation. Think of it like a balancing act – whatever you do to one side, you must do to the other to keep the equation balanced. This principle is the foundation of all equation-solving techniques.

There are primarily four basic operations we use to manipulate equations:

1. Addition: If a number is being subtracted from the variable, we add that number to both sides.
2. Subtraction: If a number is being added to the variable, we subtract that number from both sides.
3. Multiplication: If the variable is being divided by a number, we multiply both sides by that number.
4. Division: If the variable is being multiplied by a number, we divide both sides by that number.

Remember, the key is to perform the opposite operation to undo what's being done to the variable. For example, if the variable is being multiplied by 2, we divide by 2 to isolate it. By mastering these operations, you are one step closer to mastering solving equations.

When tackling equations, pay close attention to the signs (positive or negative) and make sure to apply the operations correctly to both sides. A simple mistake in arithmetic can throw off your entire solution. So, take your time, double-check your work, and remember, practice makes perfect. Solving equations is a fundamental skill in mathematics, and the more you practice, the more confident you'll become. By understanding these basics, you'll be well-equipped to tackle the examples we're about to go through and any other equation that comes your way. So, keep these principles in mind, and let's move on to solving some equations!

Solving Linear Equations: Examples and Step-by-Step Solutions

Now, let's put these principles into action by solving some linear equations. Linear equations are equations where the variable is raised to the power of 1 (like x, y, etc.). These are the most common types of equations you'll encounter, and mastering them is essential for further mathematical studies. We'll break down each problem step-by-step, so you can clearly see the process.

Equation 1: 2x = 20

Our goal here is to isolate x. Notice that x is being multiplied by 2. To undo this multiplication, we need to perform the opposite operation, which is division. So, we'll divide both sides of the equation by 2:

(2x) / 2 = 20 / 2

On the left side, the 2s cancel out, leaving us with just x. On the right side, 20 divided by 2 is 10. Therefore:

x = 10

So, the solution to the first equation is x equals 10. Easy peasy, right?

Equation 2: 6x = 42

This equation is very similar to the first one. Again, x is being multiplied by a number, this time by 6. To isolate x, we'll divide both sides of the equation by 6:

(6x) / 6 = 42 / 6

The 6s on the left side cancel out, and 42 divided by 6 is 7. So:

x = 7

Our solution for the second equation is x equals 7.

Equation 3: -7d = 15

In this equation, we have a negative coefficient (-7) multiplying the variable d. Don't let the negative sign intimidate you! We still use the same principle: divide both sides by the coefficient. So, we divide both sides by -7:

(-7d) / -7 = 15 / -7

The -7s on the left side cancel out, and 15 divided by -7 is -15/7. Therefore:

d = -15/7

So, the solution for this equation is d equals -15/7, a fraction. Remember, it's perfectly okay to have fractional solutions!

Equation 4: 23y = 6

Here, we're solving for y, and it's being multiplied by 23. To isolate y, we divide both sides of the equation by 23:

(23y) / 23 = 6 / 23

The 23s on the left cancel out, and we're left with:

y = 6/23

The solution for this equation is y equals 6/23, another fraction.

These first four equations demonstrate the basic principle of solving for a variable when it's being multiplied by a coefficient. In each case, we divided both sides of the equation by the coefficient to isolate the variable. By working through these examples, you've gained valuable practice in applying this core concept. Now, let's move on to some equations that involve division!

Solving Equations with Fractions

Now, let's tackle equations where the variable is being divided by a number. This might seem a bit trickier at first, but don't worry, the process is just as straightforward as before. Remember our golden rule: do the opposite operation to isolate the variable. In this case, since the variable is being divided, we'll use multiplication.

Equation 5: x/6 = 4

Here, x is being divided by 6. To undo this division, we need to multiply both sides of the equation by 6:

(x/6) * 6 = 4 * 6

On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just x. On the right side, 4 multiplied by 6 is 24. So:

x = 24

The solution for this equation is x equals 24. See? Not so scary!

Equation 6: r/7 = -1

In this equation, r is being divided by 7. To isolate r, we multiply both sides of the equation by 7:

(r/7) * 7 = -1 * 7

The 7s on the left side cancel out, and -1 multiplied by 7 is -7. Therefore:

r = -7

So, the solution for this equation is r equals -7.

Equation 7: -x/11 = 10

This equation has a negative sign in front of the fraction, but we can handle it! The key is to remember that -x/11 is the same as (-1 * x) / 11. So, we can think of x as being divided by 11 and then multiplied by -1. To isolate x, we need to undo both of these operations.

First, let's get rid of the division by multiplying both sides by 11:

(-x/11) * 11 = 10 * 11

This simplifies to:

-x = 110

Now, we have -x equals 110. To get x by itself, we need to get rid of the negative sign. We can do this by either multiplying or dividing both sides by -1:

-x * (-1) = 110 * (-1)

This gives us:

x = -110

So, the solution for this equation is x equals -110.

These examples demonstrate how to solve equations where the variable is being divided. By multiplying both sides by the denominator, you can effectively isolate the variable and find the solution. Now, let's tackle our final equation, which combines both division and a fraction on the other side of the equation.

Solving Equations with Fractions on Both Sides

Let's move on to our final equation, which presents a slightly different scenario. This time, we have a fraction on both sides of the equation, but don't worry, the same principles apply. The goal remains the same: isolate the variable. We'll use a combination of the techniques we've already learned to solve this one.

Equation 8: x/4 = 5/6

In this equation, x is being divided by 4. To isolate x, we need to multiply both sides of the equation by 4. This will get rid of the denominator on the left side and leave us with x by itself:

(x/4) * 4 = (5/6) * 4

On the left side, the 4s cancel out, leaving us with x. On the right side, we have (5/6) multiplied by 4. To multiply a fraction by a whole number, we can rewrite the whole number as a fraction with a denominator of 1:

(5/6) * (4/1)

Now, we multiply the numerators (5 * 4 = 20) and the denominators (6 * 1 = 6):

20/6

So, our equation now looks like this:

x = 20/6

However, we're not quite done yet! The fraction 20/6 can be simplified. Both 20 and 6 are divisible by 2. So, we divide both the numerator and the denominator by 2:

(20 / 2) / (6 / 2) = 10/3

Therefore, the simplified solution is:

x = 10/3

So, the solution for this equation is x equals 10/3, an improper fraction. We could also express this as a mixed number (3 1/3), but leaving it as an improper fraction is perfectly acceptable.

This final example demonstrates how to handle equations with fractions on both sides. By multiplying both sides by the denominator of the variable term and then simplifying the resulting fraction, you can successfully solve these types of equations. Remember to always simplify your answers as much as possible!

Conclusion: You've Got This!

Alright, guys! We've tackled a variety of equations together, from simple multiplication and division problems to those involving fractions. You've seen how to isolate the variable by using opposite operations, and you've learned the importance of simplifying your answers. Remember, the key to mastering equation-solving is practice, practice, practice!

Don't be afraid to make mistakes – they're a natural part of the learning process. The more you practice, the more comfortable you'll become with the different techniques and the more confident you'll feel in your ability to solve any equation that comes your way. So, keep practicing, keep asking questions, and most importantly, believe in yourself. You've got this!

Now that you've worked through these examples, try solving some equations on your own. You can find plenty of practice problems online or in textbooks. And if you ever get stuck, don't hesitate to ask for help from a teacher, tutor, or friend. Keep up the great work, and happy equation-solving!