Problem 38 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ 0.35 t+0.65(10-t)=0.55(10) $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{10}{3} \). It is correct upon checking.

Step 1: Simplify the Equation

Expand the terms on both sides of the equation. For the equation given, distribute the numbers outside the parentheses:\[ 0.35t + 0.65(10 - t) = 0.55(10) \] becomes \[ 0.35t + 6.5 - 0.65t = 5.5 \].

Step 2: Combine Like Terms

Combine the 't' terms on the left side of the equation:\[ 0.35t - 0.65t + 6.5 = 5.5 \] becomes \[ -0.30t + 6.5 = 5.5 \].

Step 3: Isolate the Variable Term

Subtract 6.5 from both sides to isolate the term with 't':\[ -0.30t = 5.5 - 6.5 \] simplifies to \[ -0.30t = -1 \].

Step 4: Solve for the Variable

Divide both sides by -0.30 to solve for 't':\[ t = \frac{-1}{-0.30} \] simplifies to \[ t = \frac{10}{3} \] which is approximately 3.33.

Step 5: Check the Solution

Substitute \( t = \frac{10}{3} \) back into the original equation to verify:\[ 0.35 \left(\frac{10}{3}\right) + 0.65 \left(10 - \frac{10}{3}\right) = 0.55 \times 10 \].Calculate each part:- \( 0.35 \times \frac{10}{3} \approx 1.17 \).- \( 10 - \frac{10}{3} \approx 6.67 \) implies \( 0.65 \times 6.67 \approx 4.33 \).Sum these results to check if they equal 5.5:- \( 1.17 + 4.33 = 5.5 \), confirming that the solution is correct.

Key Concepts

Combining Like Terms

Understanding like terms is crucial in solving equations efficiently. Like terms are terms that have identical variable parts. In equations, we often deal with expressions where several terms involve the same variable.
When combining like terms:

• Identify terms with the same variable.
• Simplify by adding or subtracting the coefficients.

In the equation given, we had terms like \(0.35t\) and \(-0.65t\) which both include the variable \(t\).
By combining these, \(0.35t - 0.65t\) simplifies to \(-0.30t\), bringing us a step closer to solving the equation.

Isolating Variables

Isolating the variable is a fundamental step in solving equations. This involves rearranging the equation so that the variable you're solving for is on one side.
In our equation, once the like terms are combined, the next step is to isolate \(-0.30t\).
Here's how you do it:

• Perform operations to both sides to remove constants from the side with the variable.
• Subtract or add terms as needed.

We subtracted 6.5 from both sides, effectively isolating the term with \(t\).
This simplification results in an equation \(-0.30t = -1\), focusing entirely on \(t\).
Remember, isolating the variable is about clarity, simplifying the path to a solution.

Solving Linear Equations

Linear equations are equations where the variable is raised to the power of one. The simplicity of linear equations comes from their straightforward solution path.
After isolating the variable, the goal is to find its value.
Here are the basic steps taken:

• Equations with isolated terms often involve simple division or multiplication to solve for the variable.
• This requires dividing both sides by the coefficient of the variable.

In our example, once \(-0.30t = -1\) was achieved, solving for \(t\) involved dividing both sides by \(-0.30\).
This resulted in \(t = \frac{10}{3}\) or approximately 3.33.
Checking your solution ensures accuracy, so substituting back can confirm the answer is correct.