Problem 85 - Intermediate Algebra

Question

Solve each equation. $$ -(2 t-0.71)=0.9(1.4-t) $$

Step-by-Step Solution

Answer

The solution for \( t \) is \(-0.5\).

Step 1: Distribute the negative sign

Start by addressing the left side of the equation. When you multiply the negative sign by the terms inside the parenthesis, you get:\[ -2t + 0.71 \]

Step 2: Distribute the 0.9

On the right side of the equation, apply distribution of 0.9 to both terms in the parenthesis:\[ 0.9 imes 1.4 - 0.9 imes t \]This simplifies to:\[ 1.26 - 0.9t \]

Step 3: Rewrite the equation

Combine the results of steps 1 and 2 to rewrite the equation:\[ -2t + 0.71 = 1.26 - 0.9t \]

Step 4: Combine like terms

Move all terms with \( t \) to one side and constant terms to the other side:\[ -2t + 0.9t = 1.26 - 0.71 \]Simplify this to:\[ -1.1t = 0.55 \]

Step 5: Solve for \( t \)

Now, solve for \( t \) by dividing both sides by \(-1.1\):\[ t = \frac{0.55}{-1.1} \]This simplifies to:\[ t = -0.5 \]

Key Concepts

Distributive Property

The distributive property is a core concept in algebra. It helps simplify equations by distributing a multiplier across terms inside parentheses. In the exercise example, start by distributing the negative sign in the expression \(-(2t - 0.71)\). This sign is multiplied by each part inside the parentheses:

• Multiplying \-1\ by \(2t\) gives you \(-2t\).
• Multiplying \-1\ by \(-0.71\) results in \+0.71\, because a negative times a negative is a positive.

This transforms the expression into \(-2t + 0.71\). Using the distributive property makes it easier to handle equations by working with simpler terms. In the same equation, you also apply the distributive property on the right side to distribute \0.9\ across terms within parentheses. Always remember:

• Be consistent with signs (positive or negative).
• Distribute to every term within the parentheses.

Combining Like Terms

After distribution, the next important step is to combine like terms. Like terms are those that have identical variable parts, such as \(t\) in our equation. In the equation \(-2t + 0.71 = 1.26 - 0.9t\), you want to combine the \(t\)-terms into a single term.To do this, you add or subtract the coefficients (the numbers in front of the variables). In the expression \(-2t + 0.9t\), combine these like terms by:

• Subtracting the coefficients: \(-2 + 0.9 = -1.1\).

This simplifies that side of the equation into \(-1.1t\). Like terms make it easier to navigate and solve equations because they consolidate the equation into a more manageable form.Remember:

• Identify terms with the same variable to combine them.
• Keep track of the signs of each term to combine correctly.

Isolating Variables

Isolating the variable means getting the variable by itself on one side of the equation. This makes the equation solve for that variable directly. In our example, you have the equation \(-1.1t = 0.55\).\
\>To isolate \(t\), perform the same operation on both sides of the equation to maintain the balance:

• Divide each side by the coefficient of the variable, which is \(-1.1\). So, \(t = \frac{0.55}{-1.1}\).

This makes it easier to see the solution for \(t\). Performing operations carefully means you'll arrive at the correct solution. Once isolated, the division yields \t = -0.5\.Key tips:

• Perform operations equally on both sides of the equation.
• Be cautious with negative operations; they affect the results differently than positive ones.
• Double-check each step to ensure accuracy.