Solving Equations: Step-by-Step Guide
Hey guys! Let's dive into solving some equations together. We've got six equations here, and we'll break them down step by step. Don't worry, it's easier than it looks! We'll tackle each one methodically, making sure we understand every move. So, grab your pencils, and let's get started!
1) 87 : (4x + 5) = 3
In this first equation, our main goal is to isolate 'x'. To do that, we need to undo the operations that are being applied to it. Think of it like peeling back layers of an onion until you get to the core. First, we see that 87 is being divided by the expression (4x + 5), and the result is 3. To start isolating 'x', we want to get rid of that division. How do we do that? We multiply both sides of the equation by (4x + 5).
Here's how it looks:
87 / (4x + 5) = 3
Multiply both sides by (4x + 5):
87 = 3 * (4x + 5)
Now, we need to deal with the right side of the equation. We have 3 multiplied by the expression (4x + 5). To simplify this, we use the distributive property. Remember, that means we multiply the 3 by each term inside the parentheses:
87 = 3 * 4x + 3 * 5
87 = 12x + 15
Great! Now we have a more straightforward equation. Our next step is to isolate the term with 'x' (which is 12x). To do this, we need to get rid of the +15 on the right side. How? We subtract 15 from both sides:
87 - 15 = 12x + 15 - 15
72 = 12x
We're getting closer! Now we have 12x = 72. The last step to isolate 'x' is to get rid of the 12 that's multiplying it. We do this by dividing both sides by 12:
72 / 12 = 12x / 12
6 = x
So, the solution to the first equation is x = 6. Woo-hoo! We solved it! But wait, there's more! Let's keep going.
2) 17 * (3x - 16) = 85
Alright, let's tackle the second equation: 17 * (3x - 16) = 85. Remember our goal: isolate 'x'. This time, we have 17 multiplied by the expression (3x - 16). Just like before, we'll use the distributive property to simplify things. We multiply 17 by each term inside the parentheses:
17 * 3x - 17 * 16 = 85
51x - 272 = 85
Now we have 51x - 272 = 85. To isolate the term with 'x' (51x), we need to get rid of the -272. How do we do that? We add 272 to both sides of the equation:
51x - 272 + 272 = 85 + 272
51x = 357
We're almost there! We have 51x = 357. The last step to isolate 'x' is to get rid of the 51 that's multiplying it. We do this by dividing both sides by 51:
51x / 51 = 357 / 51
x = 7
So, the solution to the second equation is x = 7. Awesome! Two down, four to go.
3) 9 * (6x - 13) = 153
Okay, equation number three: 9 * (6x - 13) = 153. You guys are getting the hang of this, I can tell! We're still on the mission to isolate 'x'. Just like before, we start by distributing the 9 across the terms inside the parentheses:
9 * 6x - 9 * 13 = 153
54x - 117 = 153
Now we have 54x - 117 = 153. To get the term with 'x' (54x) by itself, we need to get rid of the -117. What's the move? We add 117 to both sides:
54x - 117 + 117 = 153 + 117
54x = 270
Almost there! We have 54x = 270. The final step to isolate 'x' is to divide both sides by 54:
54x / 54 = 270 / 54
x = 5
So, the solution to the third equation is x = 5. Fantastic! Halfway there!
4) 3 * (8x + 51) = 201
Moving on to equation four: 3 * (8x + 51) = 201. By now, you guys probably know the drill. We want to isolate 'x', and the first thing we do is distribute the 3 across the terms in the parentheses:
3 * 8x + 3 * 51 = 201
24x + 153 = 201
Now we have 24x + 153 = 201. To get the term with 'x' (24x) alone, we need to get rid of the +153. How? Subtract 153 from both sides:
24x + 153 - 153 = 201 - 153
24x = 48
We're in the home stretch! We have 24x = 48. To isolate 'x', we divide both sides by 24:
24x / 24 = 48 / 24
x = 2
So, the solution to the fourth equation is x = 2. Great job! Just two more to go.
5) (46x - 57) : 3 = 27
Equation number five: (46x - 57) : 3 = 27. This time, we have a division right off the bat. To start isolating 'x', we need to undo that division. How do we do that? We multiply both sides of the equation by 3:
(46x - 57) / 3 * 3 = 27 * 3
46x - 57 = 81
Now we have 46x - 57 = 81. To isolate the term with 'x' (46x), we need to get rid of the -57. What's the move? We add 57 to both sides:
46x - 57 + 57 = 81 + 57
46x = 138
Almost there! We have 46x = 138. To isolate 'x', we divide both sides by 46:
46x / 46 = 138 / 46
x = 3
So, the solution to the fifth equation is x = 3. Woo-hoo! One more to go!
6) (19x + 62) : 15 = 13
Last but not least, equation number six: (19x + 62) : 15 = 13. We're finishing strong! Just like in the previous equation, we have a division to deal with first. To undo the division by 15, we multiply both sides of the equation by 15:
(19x + 62) / 15 * 15 = 13 * 15
19x + 62 = 195
Now we have 19x + 62 = 195. To isolate the term with 'x' (19x), we need to get rid of the +62. How? We subtract 62 from both sides:
19x + 62 - 62 = 195 - 62
19x = 133
We're at the final step! We have 19x = 133. To isolate 'x', we divide both sides by 19:
19x / 19 = 133 / 19
x = 7
So, the solution to the sixth equation is x = 7. We did it! We solved all six equations!
Conclusion
Great job, everyone! You've successfully tackled six equations. Remember, the key to solving equations is to isolate the variable by undoing the operations that are being applied to it. Whether it's using the distributive property, adding, subtracting, multiplying, or dividing, each step brings you closer to the solution. Keep practicing, and you'll become equation-solving pros in no time!