Problem 105 - Intermediate Algebra

Question

Solve each equation. $$ 6+4 t-1=6-15 t+12 t-8 $$

Step-by-Step Solution

Answer

The solution to the equation is \(t = -1\).

Step 1: Simplify Each Side

First, simplify the expression on each side of the equation. On the left side: \[ 6 + 4t - 1 = 5 + 4t \]On the right side: \[ 6 - 15t + 12t - 8 = -2 - 3t \] So the equation becomes: \[ 5 + 4t = -2 - 3t \]

Step 2: Move Variable Terms to One Side

Choose one side of the equation to move all the variable terms to. Let's move all terms involving \(t\) to the left side.Add \(3t\) to both sides:\[ 5 + 4t + 3t = -2 - 3t + 3t \]This simplifies to:\[ 5 + 7t = -2 \]

Step 3: Isolate the Variable Term

To solve for \(t\), you need to isolate the \(t\) term.Subtract 5 from both sides:\[ 5 + 7t - 5 = -2 - 5 \]Which simplifies to:\[ 7t = -7 \]

Step 4: Solve for the Variable

Divide both sides by 7 to solve for \(t\):\[ t = \frac{-7}{7} \]Simplifying gives:\[ t = -1 \]

Key Concepts

Simplifying Expressions

When working with linear equations, the first step is often to simplify the expressions on each side. Simplifying means changing the expression to its simplest form. You do this by combining like terms. Like terms are terms that contain the same variable raised to the same power.

For example, in the equation, terms like \(4t\) and \(-15t\) are like terms because they both contain the variable \(t\). You can combine them by adding or subtracting their coefficients (the numbers in front of the variables).

In the given exercise, we start by simplifying both sides:

• On the left side: \(6 + 4t - 1\) becomes \(5 + 4t\); we combined \(6\) and \(-1\).
• On the right side: \(6 - 15t + 12t - 8\) simplifies to \(-2 - 3t\). We combined \(-15t\) and \(12t\), as well as \(6\) and \(-8\).

These steps are crucial for preparing an equation for easy solving by making it more manageable.

Solving Equations

Once the expressions are simplified, the next task is to solve the equation. Solving equations involves finding the value of the variable that makes the equation true.

In our example, after simplification, we have \(5 + 4t = -2 - 3t\). The strategy to solve this is to get all terms containing the variable on one side of the equation and constant terms on the other side.

Here's how you continue:

• Add \(3t\) to both sides to get \(5 + 4t + 3t = -2 - 3t + 3t\), which simplifies to \(5 + 7t = -2\).

By doing this, you're effectively balancing the equation, maintaining its equality, while bringing it one step closer to isolating the variable.

Isolate the Variable

The final part of solving a linear equation involves isolating the variable. This means getting the variable by itself on one side of the equation, so that you can clearly see its value.

Continuing from our example, we now have \(5 + 7t = -2\). Here’s how to isolate \(t\):

• Subtract \(5\) from both sides to move constant terms away from the side with \(t\): \(5 + 7t - 5 = -2 - 5\) simplifies to \(7t = -7\).
• Finally, divide both sides by the coefficient of \(t\), which is \(7\): \(t = \frac{-7}{7}\), simplifying to \(t = -1\).

By isolating \(t\), we determine that \(t = -1\) satisfies the original linear equation. These steps ensure precision and clarity in reaching a solution.