Problem 77 - Beginning and Intermediate Algebra

Question

\(7 t+2(t+1)=4\)

Step-by-Step Solution

Answer

t = 2/9

Step 1: Distribute the Terms

Expand the equation by distributing the 2 into the parenthesis. This turns the equation from 7t + 2(t+1) = 4 to 7t + 2t + 2 = 4

Step 2: Combine Like Terms

Add the terms involving t together: 7t + 2t = 9t. Now you have 9t + 2 = 4.

Step 3: Isolate the Variable Term

Subtract 2 from both sides to remove the constant term from the left side. 9t + 2 - 2 = 4 - 2. This simplifies to 9t = 2.

Step 4: Solve for the Variable

Divide both sides of the equation by 9 to isolate t. 9t / 9 = 2 / 9. Thus, t = 2/9.

Key Concepts

Distributive Property

The distributive property is an important tool in algebra. Think of it like spreading out multiplication over addition.
In our exercise, the equation starts as: \[7t + 2(t + 1) = 4\]
The number 2 is outside the parentheses and needs to be multiplied by each term inside the parentheses (t and 1). This turns the equation into: \[7t + 2t + 2 = 4\]
Using the distributive property correctly helps break down more complex expressions into simpler ones, making it easier to solve the equation.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable. In our exercise, after using the distributive property, we get: \[7t + 2t + 2 = 4\]
Now, combine the terms that have the variable 't'.\[7t + 2t = 9t\]
So, the equation simplifies to: \[9t + 2 = 4\]
Combining like terms is really about simplifying the equation, making it easier to spot how to isolate the variable.

Isolating the Variable

Isolating the variable is a key step in solving algebraic equations. You want to get the unknown variable (in this case, 't') by itself on one side of the equation.
From the previous step, our equation is: \[9t + 2 = 4\]
First, remove the constant term (2) from the left side. Subtract 2 from both sides: \[9t + 2 - 2 = 4 - 2\]This simplifies to: \[9t = 2\]
Now, the variable term (9t) is isolated. The goal is to have only 't' on one side.

Solving Algebraic Equations

To finish solving the algebraic equation, you need to isolate 't' completely. From the previous step, we have: \[9t = 2\]
To isolate 't', divide both sides of the equation by 9: \[\frac{9t}{9} = \frac{2}{9}\]This simplifies to: \[t = \frac{2}{9}\]
Solving algebraic equations often involves several steps like using the distributive property, combining like terms, and isolating the variable. Practicing these techniques will help you master solving any linear equations you encounter.