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Chapter 1: Problem 61

Solve. $$ 4(t-3)-t=6 $$

Short Answer

Expert verified
t = 6

Step by step solution

01

- Distribute the 4

Distribute the 4 to both terms inside the parentheses: \[ 4(t-3)-t=6 \] becomes \[ 4t - 12 - t = 6 \]
02

- Combine like terms

Combine the terms with \( t \): \[ 4t - t - 12 = 6 \] becomes \[ 3t - 12 = 6 \]
03

- Isolate the term with \( t \)

Add 12 to both sides to isolate the term with \( t \): \[ 3t - 12 + 12 = 6 + 12 \] simplifies to \[ 3t = 18 \]
04

- Solve for \( t \)

Divide both sides by 3 to solve for \( t \): \[ \frac{3t}{3} = \frac{18}{3} \] simplifies to \[ t = 6 \]

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Key Concepts

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

distributive property
To start solving this problem, we use the distributive property. This property helps in expanding expressions that involve multiplication over addition or subtraction. For example, in the equation given, we have to distribute the 4 to both terms inside the parentheses \(t-3\). This means we multiply 4 by both \(t\) and -3. So, \[4(t-3) = 4 \cdot t - 4 \cdot 3 = 4t - 12 \] The expression inside the parentheses is now simplified, making the equation easier to work with.
combining like terms
Once we have used the distributive property, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this exercise, after distributing, the equation becomes \[4t - 12 - t = 6 \] Now, we combine the terms with the variable \(t\). The terms are \(4t\) and \(-t\). Adding them, we get: \[4t - t = 3t \] So, the equation now looks like: \[3t - 12 = 6 \] Combining like terms simplifies the equation and makes the next steps easier.
isolating variables
In the next step, our goal is to isolate the variable \(t\) on one side of the equation. Currently, we have: \[3t - 12 = 6 \] We need to get rid of the \(-12\) on the left side. To do this, we add 12 to both sides: \[3t - 12 + 12 = 6 + 12 \] Simplifying this, we get: \[3t = 18 \] Now, the variable term \(3t\) is isolated, making it possible to solve for \(t\).
solving for a variable
The final step is to solve for the variable \(t\). The equation we now have is: \[3t = 18 \] To find the value of \(t\), we need to divide both sides by 3: \[\frac{3t}{3} = \frac{18}{3} \] This simplifies to: \[t = 6 \] So, the value of the variable \(t\) is 6. By following these steps 鈥� using the distributive property, combining like terms, isolating the variable, and solving for it 鈥� we have successfully determined that \(t = 6\).

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Most popular questions from this chapter

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ \mathrm{H} \subseteq R $$

Calculate using the rules for order of operations. $$ 15-1+\sqrt{5^{2}-(3+1)^{2}}(-1) $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 43-(-9+2)^{2}+18 \div 6 \cdot(-2) $$

Write an equivalent expression using the distributive law. $$ 6(s-t) $$

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