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In mathematics, equations are like puzzles. Solving equations means finding the mystery number that makes the equation true. When we have an equation such as \[ -33 = 5t + 2 \]we are given a problem where the goal is to discover the value of \( t \). This value, when placed back into the equation, should make both sides equal. This process is important because it allows us to understand the relationships and properties of numbers and symbols in algebra. By learning to solve different types of equations, we build a foundation for more complex mathematical concepts. This equation is called a linear equation, which means it forms a straight line when graphed. Working through each step to find the solution helps build confidence and problem-solving skills.

Variable isolation is a key step in solving equations. It means getting the variable, in this case, \( t \), by itself on one side of the equation. This helps us find its value.

• Step 1: Move constants. We first need to remove numbers that are not connected to our variable. In the original equation,\[ -33 = 5t + 2 \], we subtract 2 from both sides. This step isolates the variable term by itself with its coefficient:\[ -33 - 2 = 5t \]This simplifies to:\[ -35 = 5t \]
• Step 2: Remove the coefficient. After isolating the variable term, we need to deal with the multiplier or coefficient of the variable, which is 5 in this case. We can divide both sides of the equation by 5 to make \( t \) stand alone:\[ \frac{-35}{5} = t \]
  This simplifies and results in:\[ -7 = t \]

By successfully isolating the variable, the mystery number \( t \) is found. This is why isolating the variable is such a critical element of solving equations.

Following a step-by-step solution approach is beneficial, especially when learning to solve linear equations. This methodical process ensures that you do not miss any important steps and keeps the process clear and organized.
Your steps are:1. **Identify the equation structure:** Look at what needs to be isolated. Our variable, \( t \), is mixed in with other numbers.2. **Isolate the variable with subtraction:** Using subtraction helps to remove extra numbers. After subtracting 2, our equation looks like:\[ -35 = 5t \]This stage essentially clears any extra numbers away from the \( t \) term.3. **Divide to simplify:** The division is the last step to make sure that \( t \) is all by itself. By dividing both sides by the coefficient 5, we solve for \( t \):\[ t = -7 \]This approach promotes clarity and ensures that solving equations becomes a manageable task, taking things one step at a time. This is crucial for developing a deep understanding of algebra and helps in tackling more challenging mathematical problems effectively.