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Chapter 5: Problem 93

Solve and check. $$10 y-3 y=21$$

Short Answer

Expert verified
The solution to the equation \( 10y - 3y = 21 \) is \( y = 3 \).

Step by step solution

01

Simplify Left Side

First, simplify the left side of the equation by combining like terms. So, \( 10y - 3y \) simplifies to \( 7y \). So the equation now becomes \( 7y = 21 \).
02

Solve for y

To isolate \( y \), divide both sides of the equation by 7. This gives: \( y = 21 / 7 = 3 \).
03

Check the Solution

Substitute the found value for \( y \) back into the original equation to verify the solution. So, \( 10*3 - 3*3 = 21 \) simplifies to \(30 - 9 = 21\), which confirms that our solution is correct.

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Key Concepts

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

Combining Like Terms
When solving linear equations, one of the first steps is to identify and combine like terms. Like terms are terms in an equation that have the same variable raised to the same power. In our given equation, \(10y - 3y = 21\), both \(10y\) and \(3y\) are like terms because they both contain the variable \(y\).
By combining them, you simplify the equation and make it easier to solve. Here, you subtract \(3y\) from \(10y\), resulting in \(7y\).
  • This process can reduce the complexity of your problem.
  • It also helps you see the structure of the equation more clearly.
Once you've combined like terms, you're well on your way to solving the equation.
Isolating the Variable
Isolating the variable is crucial in solving equations. This means getting the variable, in this case \(y\), alone on one side of the equation. After combining like terms, we have \(7y = 21\).
To isolate \(y\), divide both sides by the coefficient of \(y\), which is 7. This kind of step works because it uses the principle of equivalence: performing the same operation on both sides does not change the equation's solution.
You end up with
  • the equation \(y = 3\).
  • This tells us the value of \(y\) that makes our equation true.
Mastering the art of isolating variables is a key skill in algebra.
Verifying Solutions
Verification is the final step to ensure that your solution is correct. It also provides a confidence check for your work. After finding \(y = 3\), substitute it back into the original equation to see if it holds true.
Taking the original equation \(10y - 3y = 21\), replace \(y\) with 3, resulting in \(10 \times 3 - 3 \times 3 = 21\).
Simplify the arithmetic
  • \(30 - 9 = 21\)
which confirms that our solution is accurate since both sides equal each other.
This step is essential because it catches any potential errors in your calculations, solidifying your understanding of the equation.

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