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Chapter 2: Problem 61

Solve each equation, and check your solution. $$ (5 y+6)-(3+4 y)=10 $$

Short Answer

Expert verified
y = 7

Step by step solution

01

- Simplify the equation

Combine like terms on the left side of the equation. We have (5y + 6) - (3 + 4y) = 10.Remove the parentheses: 5y + 6 - 3 - 4y = 10.
02

- Combine like terms

Combine the terms involving y and the constant terms:(5y - 4y) + (6 - 3) = 10,which simplifies to: y + 3 = 10.
03

- Isolate the variable

Subtract 3 from both sides to isolate y:y + 3 - 3 = 10 - 3,which simplifies to: y = 7.
04

- Check the solution

Substitute y = 7 back into the original equation to verify the solution:(5(7) + 6) - (3 + 4(7)) = 10.Calculate each part:(35 + 6) - (3 + 28) = 10,which simplifies to: 41 - 31 = 10.Since both sides of the equation are equal, y = 7 is the correct solution.

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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 combine like terms. Like terms are terms that have the same variable raised to the same power. In the given equation, (5y + 6) - (3 + 4y) = 10, we need to remove the parentheses and combine similar terms. To do this:
  • First, distribute any signs outside the parentheses across the terms inside. For instance, -(3 + 4y) becomes -3 - 4y.
  • Remove parentheses to get: 5y + 6 - 3 - 4y = 10.
  • Combine the y terms: 5y - 4y, which simplifies to y.
  • Next, combine the constants: 6 - 3, which simplifies to 3.
After combining, we have the simplified equation: y + 3 = 10.
Isolating the Variable
After combining like terms, the next key step is to isolate the variable to solve for it. In our simplified equation, y + 3 = 10, we want to isolate y:
  • Subtract 3 from both sides of the equation to keep it balanced. This gives us: y + 3 - 3 = 10 - 3.
  • Simplify the equation to get y = 7.
By isolating the variable, we have found the value of y. It's important to perform the same operation on both sides of the equation to maintain the balance. This helps us solve for the unknown variable accurately.
Checking Solutions
The final step in solving linear equations is checking your solution. This ensures that your answer is correct. To verify the solution y = 7:
  • Substitute y = 7 back into the original equation: (5y + 6) - (3 + 4y) = 10 becomes (5(7) + 6) - (3 + 4(7)) = 10.
  • Calculate each part: 5(7) = 35 and 4(7) = 28.
  • Substitution gives us: (35 + 6) - (3 + 28) = 10, or 41 - 31 = 10.
  • Since 10 equals 10, both sides of the equation are equal, confirming that y = 7 is the correct solution.
Always substitute your solution back into the original equation to ensure accuracy. This step verifies that no mistakes were made during manipulation and that the solution satisfies the original equation.

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