MC
Okay, so I need to figure out how to solve this problem where I have to find the value of 'a' in the equation 2a + 5 = 17. Let me start by recalling some basic algebra. The goal here is to isolate the variable 'a' on one side of the equation.
First, I remember that to solve for a variable, I should perform inverse operations to get rid of the constants attached to it. In this case, the equation is 2a + 5 = 17. The left side has 2a plus 5, so I need to undo the addition of 5 first. The inverse of addition is subtraction, so I should subtract 5 from both sides to keep the equation balanced. Let me write that down:
2a + 5 - 5 = 17 - 5
Simplifying both sides, the +5 and -5 on the left cancel each other out, leaving just 2a. On the right side, 17 minus 5 is 12. So now the equation is:
2a = 12
Now, I need to get rid of the coefficient 2 that's multiplied by 'a'. The inverse of multiplication is division, so I should divide both sides by 2. Let me do that:
2a / 2 = 12 / 2
Simplifying that, the 2's on the left cancel out, leaving just 'a'. On the right, 12 divided by 2 is 6. So:
a = 6
Wait, let me check if that makes sense. If I plug a = 6 back into the original equation, does it hold true?
Original equation: 2a + 5 = 17
Substituting a with 6: 2*6 + 5 = 12 + 5 = 17
Yes, that equals 17, which matches the right side of the equation. So that checks out. Therefore, a = 6 is the correct solution.
But wait, let me make sure I didn't make any mistakes in my steps. Sometimes when you're first learning, it's easy to mix up the order of operations or forget to apply the inverse operation to both sides. Let me go through it again step by step.
Starting with 2a + 5 = 17. The first step is to subtract 5 from both sides. That gives 2a = 12. Then divide both sides by 2, resulting in a = 6. Plugging back in, 2*6 is 12, plus 5 is 17. Yep, that's correct.
Is there another way to approach this problem? Maybe by dividing first? Let me see. If I divide the entire equation by 2 first, it would be (2a + 5)/2 = 17/2. That simplifies to a + 2.5 = 8.5. Then subtract 2.5 from both sides: a = 8.5 - 2.5 = 6. So same result. So either method works.
But typically, the standard method is to reverse the order of operations, which is undo addition before multiplication. That's because of PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). So when solving equations, you usually do the opposite: reverse the order, so deal with addition/subtraction first, then multiplication/division.
Another thing to consider: if the equation had variables on both sides, you'd need to collect like terms first. But in this case, it's straightforward.
https://www.18122.com/url/usdt/usdt1156.JPEG
Let me also think about common mistakes. A student might forget to subtract 5 from both sides and instead subtract 5 only from one side, leading to an incorrect equation. For example, if someone writes 2a = 17 - 5, that's correct, but if they write 2a = 17 and then subtract 5, that's wrong. So it's crucial to subtract 5 from both sides.
Or another mistake could be dividing by 2 incorrectly. For instance, if someone divides 17 by 2 first, getting 8.5, and then subtracts 5, but that's not the right approach. Wait, actually, that's the second method I tried earlier, which does work. So as long as you apply the operations correctly to both sides, the order can vary.
But the key takeaway is that whatever you do to one side, you must do to the other to maintain the equality. That's the fundamental principle of algebra.
Let me also visualize this. Imagine a balance scale. On one side, you have 2a + 5, and on the other, 17. To find the value of 'a', you need to remove weights from both sides equally. Removing 5 from both sides leaves 2a on one side and 12 on the other. Then, splitting the 2a into two equal parts (dividing by 2) gives 'a', and the 12 split into two parts is 6. So the scale balances when 'a' is 6. That's a helpful way to think about it.
Alternatively, using algebra tiles: you have two tiles of 'a' and five unit tiles on one side, and seventeen unit tiles on the other. To solve, you remove five unit tiles from both sides, leaving two 'a' tiles equal to twelve unit tiles. Then each 'a' tile is equivalent to six unit tiles.
I think I've covered different methods and checked the solution multiple times. It seems solid. The answer is definitely 6. But just to be absolutely thorough, let me try another example to test my understanding. Suppose the equation was 3a + 4 = 19. Following the same steps: subtract 4 from both sides to get 3a = 15, then divide by 3 to get a = 5. Plugging back in: 3*5 +4 = 15 +4 =19. Correct. So the method works here too.
Therefore, confident that the solution a = 6 is correct for the original equation.
To solve the equation (2a + 5 = 17) for (a), follow these steps:
Subtract 5 from both sides to isolate the term with (a):
[
2a + 5 - 5 = 17 - 5 \implies 2a = 12
]
Divide both sides by 2 to solve for (a):
[
\frac{2a}{2} = \frac{12}{2} \implies a = 6
]
Verification:
Substitute (a = 6) back into the original equation:
[
2(6) + 5 = 12 + 5 = 17
]
The left side equals the right side, confirming the solution is correct.
Answer: (\boxed{6})
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