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Chapter 2: Problem 52

Solve each equation, and check the solution. \(8(t-3)+4 t=6(2 t+1)-10\)

Short Answer

Expert verified
The equation has no solution.

Step by step solution

01

Distribute on both sides

First, distribute the numbers outside the parentheses on both sides of the equation. On the left side, distribute 8 to both terms inside the parentheses: 8(t-3) + 4t = 8t - 24 + 4t On the right side, distribute 6 to both terms inside the parentheses: 6(2t+1) - 10 = 12t + 6 - 10
02

Combine like terms

Next, combine the like terms on each side of the equation: On the left side: 8t - 24 + 4t = 12t - 24 On the right side: 12t + 6 - 10 = 12t - 4
03

Isolate variable terms

Move all variable terms to one side and constant terms to the other side. Subtract 12t from both sides: 12t - 24 = 12t - 4 12t - 12t - 24 = 12t - 12t - 4 -24 = -4
04

Simplify the equation

As -24 does not equal -4, there is a contradiction. Therefore, there is no solution to this equation.

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Key Concepts

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

algebraic equations
Algebraic equations are mathematical statements that show the equality of two expressions. They often include variables, numbers, and operations like addition, subtraction, multiplication, and division. Solving these equations means finding the value of the variable that makes the equation true.
In the given exercise, we have an equation with the variable t: \[8(t-3) + 4t = 6(2t+1) -10\]
The goal is to determine the value of t that satisfies this equation.
distributive property
The Distributive Property is a fundamental principle in algebra that allows us to multiply a single term by each term inside a parenthesis. It states that \(a(b + c) = ab + ac\).
Let鈥檚 apply it to the given equation:
  • On the left side: \(8(t-3) + 4t\)
  • Distribute 8: \(8t - 24\)
    Now we have \(8t - 24 + 4t\).
  • On the right side: \(6(2t+1) - 10\)
  • Distribute 6: \(12t + 6\)
    Resulting in \(12t + 6 - 10\).
combining like terms
Combining like terms simply means adding or subtracting terms that contain the same variable raised to the same power. This helps simplify the equation.
From our distributed terms:
  • On the left side: \(8t - 24 + 4t\)
  • Combine like terms: \(8t + 4t - 24\), which gives \(12t - 24\).
  • On the right side: \(12t + 6 - 10\)
  • Combine like terms: \(12t - 4\).

Our equation is now \(12t - 24 = 12t - 4\).
no solution scenarios
In some algebraic equations, you might encounter a situation where there is no possible value for the variable that satisfies the equation. This typically happens when you simplify the equation and end up with a false statement.
After combining like terms, we had: \(12t - 24 = 12t - 4\).
Subtracting \(12t\) from both sides, we get \(-24 = -4\). This is a contradiction because \(-24\) does not equal \(-4\).
Therefore, there is no solution to this equation. This is a classic example of a 'no solution' scenario, often encountered when the same variable terms appear with different constants on both sides of the equation.

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