Problem 39 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ -3(2 t-5)+2 t=5 t-4 $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{19}{9} \).

Step 1: Distribute the Negative Sign

First, distribute the \(-3\) across the expression inside the parentheses. This means you multiply \(-3\) by \(2t\) and by \(-5\). This results in \(-3 \times 2t = -6t\) and \(-3 \times (-5) = 15\). The equation now becomes \(-6t + 15 + 2t = 5t - 4\)."

Step 2: Combine Like Terms on the Left

Now, simplify the left side of the equation by combining like terms \(-6t\) and \(+2t\). This gives you \(-4t\), so the equation becomes \(-4t + 15 = 5t - 4\)."

Step 3: Move Variable Terms to One Side

Next, add \(+4t\) to both sides of the equation to move all terms with \(t\) to the right side. This results in \(+15 = 9t - 4\)."

Step 4: Isolate the Variable Term

To isolate the term with \(t\), add \(+4\) to both sides to help remove the constant term from the right side. This gives you \(+19 = 9t\)."

Step 5: Solve for the Variable

Finally, divide both sides by \(9\) to solve for \(t\). This results in \((19/9) = t\), so \(t = \frac{19}{9}\).

Key Concepts

Distributive Property

When working with linear equations, you often encounter expressions with parentheses. The distributive property helps you simplify such expressions. It involves multiplying each term inside the parentheses by the factor outside. In our exercise, the expression inside the parentheses is \(2t - 5\), and the factor is \(-3\). This means:

• Multiply \(-3\) by \(2t\) resulting in \(-6t\)
• Multiply \(-3\) by \(-5\) resulting in \(15\)

Applying the distributive property changes the equation to \(-6t + 15 + 2t = 5t - 4\). Breaking down complex expressions this way makes the equation easier to solve in subsequent steps.

Combining Like Terms

After using the distributive property, you are left with terms that might need further simplification. Combining like terms means putting together terms that have the same variable. For example, in the equation \(-6t + 15 + 2t\), \(-6t\) and \(+2t\) are like terms because they both involve \(t\). By adding these terms together, you get \(-4t\).Like terms are combined by performing basic addition or subtraction. The transformation leads to a simplified version of the equation: \(-4t + 15 = 5t - 4\). This step ensures the equation is easier to manage.

Isolating Variables

Isolating the variable means maneuvering the equation so that the variable you are solving for is on one side. For our equation, this involves moving all terms with \(t\) to the same side. To do this, add \(+4t\) to both sides,
resulting in \(15 = 9t - 4\). This extracts \(t\) terms to one side and prepares the equation for the next stage of simplification.By strategically moving terms containing the variable, you set up the equation to finally solve for the variable itself, ensuring a key foundational step in equation-solving.

Equation Solving Steps

Solving a linear equation is a methodical process that requires several key steps. Here’s a breakdown:

• Apply the distributive property - Simplify expressions with parentheses
• Combine like terms - Simplify the equation further by adding or subtracting terms with the same variable
• Isolate the variable - Adjust the equation to have the variable on one side, simplifying the equation
• Solve for the variable - Complete the final calculations to find the value of the variable

In our problem, after isolating, add \(+4\) to both sides to get \(19 = 9t\). Finally, solve for \(t\) by dividing both sides by \(9\): \(t = \frac{19}{9}\). These structured steps provide a clear path from problem to solution in linear equations.