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Chapter 2: Problem 11

Solve and check. Label any contradictions or identities. $$7 t-8=27$$

Short Answer

Expert verified
The solution is \( t = 5 \). Verification confirms this.

Step by step solution

01

Isolate the variable term

To isolate the term containing the variable, add 8 to both sides of the equation:\[ 7t - 8 + 8 = 27 + 8 \]This simplifies to:\[ 7t = 35 \]
02

Solve for the variable

Now, divide both sides by 7 to solve for \( t \):\[ \frac{7t}{7} = \frac{35}{7} \]This simplifies to:\[ t = 5 \]
03

Verify the solution

Substitute \( t = 5 \) back into the original equation to verify:\[ 7(5) - 8 = 27 \]This simplifies to:\[ 35 - 8 = 27 \]Since both sides of the equation are equal, the solution is verified.

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Key Concepts

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

isolating variable
To solve linear equations, the first crucial step is isolating the variable. This means getting the variable, like \( t \), alone on one side of the equation. To achieve this, you need to perform operations that cancel out other numbers surrounding the variable. In the exercise, the initial equation is \(7t - 8 = 27\). Here are the steps to isolate the variable:

  • Identify the term that must be moved to isolate the variable. In this case, it's the \(-8\) next to \(7t\).
  • Perform the opposite operation to remove the term. Since the term is \(-8\), you add \(8\) to both sides of the equation to cancel it out:
    • \[ 7t - 8 + 8 = 27 + 8 \]
  • This simplifies to \( 7t = 35 \), effectively isolating the term with the variable.

By isolating the main term, solving the equation becomes a manageable task. This step is fundamental to solving any algebraic equation.
verifying solutions
After solving the equation, it's essential to ensure your solution is correct. This process is called verifying solutions. In simple terms, it involves substituting the found value back into the original equation to see if it holds true. For our equation \(7t - 8 = 27\), we found \(t = 5\). Now let's verify:

  • Substitute \(t = 5\) back into the equation: \[ 7(5) - 8 = 27 \]
  • Simplify the left side: \[ 35 - 8 = 27 \]
  • Since the left side equals the right side, the solution is verified.

Verifying solutions is a crucial part of solving algebraic equations as it ensures you haven't made any errors. This step provides confidence that your solution is accurate and complete.
algebraic equations
Algebraic equations are mathematical statements that show the equality of two expressions. They include variables, constants, and operations like addition and multiplication. The goal is often to find the value of the variable that makes the equation true.

In the example \(7t - 8 = 27\), the equation contains:
  • A variable term \( 7t\)
  • A constant \(-8\)
  • A result \(27\)

Solving algebraic equations usually involves:
  • Isolating the variable term(s)
  • Performing inverse operations to simplify
  • Checking the solution by substitution

Through clear steps, algebraic equations can be tackled to find the unknown with accuracy.

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Most popular questions from this chapter

Match the sentence with one of the following: $$ a

Use an inequality and the five-step process to solve each problem. One side of a triangle is \(2 \mathrm{cm}\) shorter than the base. The other side is \(3 \mathrm{cm}\) longer than the base. What lengths of the base will allow the perimeter to be greater than \(19 \mathrm{cm} ?\)

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