Problem 19 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -5(3-4 t)=65 $$

Step-by-Step Solution

Answer

The solution is \(t = 4\).

Step 1: Distribute the -5

Start by distributing the -5 across the terms inside the parentheses. Multiply -5 by each term: \[ -5(3 - 4t) = -5 \times 3 + (-5) \times (-4t) = -15 + 20t \]

Step 2: Set the equation equal to 65

Substitute the expression from Step 1 into the equation: \[ -15 + 20t = 65 \]

Step 3: Add 15 to both sides

To isolate the term with \(t\), add 15 to both sides of the equation:\[ -15 + 15 + 20t = 65 + 15 \]\[ 20t = 80 \]

Step 4: Divide by 20

To solve for \(t\), divide both sides by 20:\[ \frac{20t}{20} = \frac{80}{20} \]\[ t = 4 \]

Step 5: Check the solution

Substitute \(t = 4\) back into the original equation to verify:\[ -5(3 - 4(4)) = 65 \]Calculate inside the parentheses first:\[ 3 - 16 = -13 \]Then multiply:\[ -5 \times (-13) = 65 \]Since both sides are equal, \(t = 4\) is indeed a correct solution.

Key Concepts

Solving Equations

Equations are mathematical statements that show the equality between two expressions with unknown values represented by variables. In this exercise, we had a single-variable equation involving the variable \(t\):
-5(3 - 4t) = 65.
The task was to find the value of \(t\) that makes both sides of the equation equal.
To solve an equation, you generally want to isolate the variable you're solving for. Here’s a step-by-step approach we used:

• First, distribute the constant in front of the parenthesis, which simplifies the equation.
• Next, isolate the variable by simplifying each side. This can involve combining like terms or moving numbers from one side to the other.
• Finally, solve for the variable. In this example, we divided both sides by a coefficient to finally solve for \(t\).

Solving equations is like unwrapping a present. You remove layers to reveal what's inside. Each operation you perform simplifies one layer, getting you closer to the answer.

Distributive Property

The distributive property is a crucial algebraic rule used to multiply a single term across terms inside a parenthesis. Understanding and applying this property effectively helps simplify equations, making them more manageable.
In this example, the distributive property was used in step 1:
\[-5(3 - 4t) = -5 \times 3 + (-5) \times (-4t) = -15 + 20t.\]This property states that multiplying a sum (or difference) by a number is the same as multiplying each term within the parenthesis by that number, then adding (or subtracting) the results.

• Think of it as distributing or spreading the multiplication over each term inside the parentheses.
• It's particularly handy in breaking down expressions that are not directly solvable without simplification.

Mastering the distributive property allows you to confidently handle equations with variables and constants bundled together in parentheses.

Verification of Solutions

Once you solve an equation, it’s important to verify that your solution is correct. This involves substituting your solution back into the original equation to check if both sides are equal.
In our example, after finding \(t = 4\), we substituted it back into the original equation:
\[-5(3 - 4(4)) = 65.\]Here's how you verify:

• Replace the variable in the original equation with your solution.
• Simplify the expression to confirm that the left side equals the right side of the equation.

In the verification process:

• We first calculated \(3 - 16 = -13\).
• Then, multiplied \(-5 \times (-13)\) which indeed equaled 65.

This confirmed that our solution was correct. Verification is like a warranty for your math work. It ensures your answer holds true and withstands scrutiny, allowing you to be confident in your results.