Solving Equations Step-by-Step Guide

by Scholario Team 37 views

Hey guys! 👋 Today, we're diving into the exciting world of solving mathematical equations. Don't worry, it's not as intimidating as it sounds! We'll break down each problem step by step, making sure you understand the logic and reasoning behind every move. Let's get started and make math a little less mysterious, shall we?

1) Solving the equation 2(2x - 3) = 6 + x

So, we're kicking things off with the equation 2(2x - 3) = 6 + x. The key here is to untangle the equation, bit by bit, until we isolate 'x' on one side. Think of it like peeling an onion – layer by layer!

Step 1: Distribute the 2

The first thing we need to do is get rid of those parentheses. We do this by distributing the '2' across the terms inside the parentheses. This means multiplying '2' by both '2x' and '-3'.

  • 2 * 2x = 4x
  • 2 * -3 = -6

So, our equation now looks like this: 4x - 6 = 6 + x

Step 2: Gather the 'x' terms

Now, let's get all the 'x' terms onto one side of the equation. The goal is to have 'x' only on one side. We can do this by subtracting 'x' from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. It's like a mathematical seesaw!

  • 4x - 6 - x = 6 + x - x
  • This simplifies to: 3x - 6 = 6

Step 3: Isolate the 'x' term

Next, we want to get the 'x' term by itself on one side. We can do this by adding '6' to both sides of the equation. This will cancel out the '-6' on the left side.

  • 3x - 6 + 6 = 6 + 6
  • This simplifies to: 3x = 12

Step 4: Solve for 'x'

Finally, we're in the home stretch! To solve for 'x', we need to get rid of the '3' that's multiplying it. We do this by dividing both sides of the equation by '3'.

  • 3x / 3 = 12 / 3
  • This gives us: x = 4

Therefore, the solution to the equation 2(2x - 3) = 6 + x is x = 4.

2) Tackling the equation 3x + 6(x + 1) = 3(x + 1) + 5

Alright, let's jump into our next equation: 3x + 6(x + 1) = 3(x + 1) + 5. This one looks a little more involved, but don't sweat it! We'll use the same principles as before – distributing, combining like terms, and isolating 'x'. Think of it as a puzzle – each step brings us closer to the solution. The more familiar you get with the process, the easier it will be to solve for 'x'. Keep in mind that mathematics is a skill that is developed with practice.

Step 1: Distribute where necessary

First, we need to deal with those parentheses. We'll distribute the '6' in '6(x + 1)' and the '3' in '3(x + 1)'.

  • 6 * x = 6x
  • 6 * 1 = 6
  • 3 * x = 3x
  • 3 * 1 = 3

Our equation now becomes: 3x + 6x + 6 = 3x + 3 + 5

Step 2: Combine like terms on each side

Next, let's simplify each side of the equation by combining like terms. On the left side, we have '3x' and '6x', which we can combine. On the right side, we have '3' and '5'.

  • 3x + 6x = 9x
  • 3 + 5 = 8

So, our equation simplifies to: 9x + 6 = 3x + 8

Step 3: Gather the 'x' terms on one side

Now, let's get all the 'x' terms onto one side of the equation. We can do this by subtracting '3x' from both sides.

  • 9x + 6 - 3x = 3x + 8 - 3x
  • This simplifies to: 6x + 6 = 8

Step 4: Isolate the 'x' term

We want to isolate the 'x' term, so let's subtract '6' from both sides of the equation.

  • 6x + 6 - 6 = 8 - 6
  • This simplifies to: 6x = 2

Step 5: Solve for 'x'

Finally, to solve for 'x', we divide both sides of the equation by '6'.

  • 6x / 6 = 2 / 6
  • This gives us: x = 1/3

Therefore, the solution to the equation 3x + 6(x + 1) = 3(x + 1) + 5 is x = 1/3.

3) Unraveling the equation 2y + 3(2y - 5) + 4 = y + 3(2y - 2) - 6

Moving on, we've got the equation 2y + 3(2y - 5) + 4 = y + 3(2y - 2) - 6. Don't be intimidated by the 'y' – it's the same process as solving for 'x'! We'll distribute, combine like terms, and isolate 'y'. Think of 'y' as just another variable waiting to be found.

Step 1: Distribute where necessary

Let's start by distributing the '3' in both '3(2y - 5)' and '3(2y - 2)'.

  • 3 * 2y = 6y
  • 3 * -5 = -15
  • 3 * 2y = 6y
  • 3 * -2 = -6

Our equation now looks like this: 2y + 6y - 15 + 4 = y + 6y - 6 - 6

Step 2: Combine like terms on each side

Now, let's combine the like terms on each side of the equation. On the left side, we have '2y' and '6y', and '-15' and '4'. On the right side, we have 'y' and '6y', and '-6' and '-6'.

  • 2y + 6y = 8y
  • -15 + 4 = -11
  • y + 6y = 7y
  • -6 - 6 = -12

So, our equation simplifies to: 8y - 11 = 7y - 12

Step 3: Gather the 'y' terms on one side

Let's get all the 'y' terms on one side by subtracting '7y' from both sides.

  • 8y - 11 - 7y = 7y - 12 - 7y
  • This simplifies to: y - 11 = -12

Step 4: Isolate the 'y' term

To isolate 'y', we'll add '11' to both sides of the equation.

  • y - 11 + 11 = -12 + 11
  • This simplifies to: y = -1

Therefore, the solution to the equation 2y + 3(2y - 5) + 4 = y + 3(2y - 2) - 6 is y = -1.

4) Deciphering the equation -3x + 18 = 4x - 2 - 3x + 18 = -x(13 - 10) + 4x - 2

Last but not least, we have a slightly complex looking equation: -3x + 18 = 4x - 2 - 3x + 18 = -x(13 - 10) + 4x - 2. Notice that this equation seems to have two parts connected by an equals sign. Actually, the heart of this equation lies in simplifying both the left and right-hand sides, and then using those simplifications to find the value of 'x'. We'll still apply the same techniques: simplifying, and bringing 'x' to one side.

Step 1: Simplify Both Sides of the Equation

First, let's focus on simplifying each side independently before trying to solve for 'x'.

Left Side: -3x + 18

This side is actually already pretty simple! There are no like terms to combine, so we'll leave it as is.

Right Side: -x(13 - 10) + 4x - 2

  • First, simplify inside the parentheses: 13 - 10 = 3
  • Now we have: -x(3) + 4x - 2
  • Multiply -x by 3: -3x + 4x - 2
  • Combine like terms (-3x and 4x): x - 2

Now that we've simplified both sides, our equation looks like this: -3x + 18 = x - 2

Step 2: Gather 'x' terms on one side

To bring all the 'x' terms to one side, we'll add '3x' to both sides of the equation. This will eliminate the '-3x' term on the left.

  • -3x + 18 + 3x = x - 2 + 3x
  • Simplifies to: 18 = 4x - 2

Step 3: Isolate the term with 'x'

We want the term with 'x' by itself, so we'll add '2' to both sides of the equation.

  • 18 + 2 = 4x - 2 + 2
  • This gives us: 20 = 4x

Step 4: Solve for 'x'

Finally, to find 'x', we'll divide both sides of the equation by '4'.

  • 20 / 4 = 4x / 4
  • This simplifies to: 5 = x or x = 5

Therefore, the solution to the equation -3x + 18 = 4x - 2 - 3x + 18 = -x(13 - 10) + 4x - 2 is x = 5.

Conclusion

And there you have it! We've successfully tackled four different equations, step by step. Remember, the key to solving equations is to take it slow, break it down, and stay organized. With practice, you'll become a math whiz in no time! Keep practicing, and don't be afraid to ask for help when you need it. You got this! 💪