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Chapter 2: Problem 29

Solve each equation. $$ 3 x+2 x=50 $$

Short Answer

Expert verified
The solution is \(x = 10\).

Step by step solution

01

Combine Like Terms

First, combine the terms with the variable. In this equation, you have \(3x\) and \(2x\), which can be combined to make \(5x\). The equation now looks like this: \(5x = 50\).
02

Isolate the Variable

Next, you want to solve for \(x\) by isolating the variable. Since \(5x = 50\), divide each side of the equation by 5 to get \(x = 10\).
03

Verify the Solution

To ensure the solution is correct, substitute \(x = 10\) back into the original equation. The equation is \(3(10) + 2(10) = 50\), which simplifies to \(30 + 20 = 50\). Since the equation holds true, our solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
In algebra, combining like terms is a crucial step toward simplifying equations. Think of it as tidying up your workspace. When you have terms that look the same, like those with the same variable and exponent, you can combine them.
This makes the equation simpler and easier to work with. Let's look at our exercise:
  • We have the terms: \(3x\) and \(2x\).
  • Both terms have an 'x', which means they are like terms.
  • You can add these together to form \(5x\).
By combining these like terms, you have simplified the equation into \(5x = 50\). It鈥檚 a way of creating a cleaner expression that speaks the same language all the way across. Remember, combining like terms is about uniting what's alike, which is like putting all coins of the same denomination in one pile.
Isolating the Variable
Once we've combined like terms, the next step in solving an algebra equation is isolating the variable. This means getting the variable alone on one side of the equation to solve for its value. Here's how you can think of it:
  • You have the equation \(5x = 50\).
  • To isolate \(x\), divide both sides of the equation by 5.
  • This calculation gives you \(x = 10\).
Isolating the variable transforms the equation into something like "x = (some number)". Each action you take should be balanced, meaning whatever you do to one side of the equation, you do to the other. This keeps the equation true, just like keeping a seesaw balanced!
Verifying the Solution
Verifying the solution is a final check to ensure the solution works in the original equation. It confirms everything you've done is correct. Let's explore how to do this:
  • Substitute \(x = 10\) back into the original equation \(3x + 2x = 50\).
  • This becomes \(3(10) + 2(10) = 50\).
  • Simplify this to \(30 + 20 = 50\), which holds true.
When both sides of the equation are equal after substitution, your solution is verified. Verification acts like a double-check, similar to reviewing a completed task to make sure everything adds up correctly. It ensures that you confidently arrive at the right solution every time.

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