Problem 39 - Prealgebra

Question

Solve each equation. Check your solution. $$1-3 y+y=5$$

Step-by-Step Solution

Answer

The solution is \(y = -2\).

Step 1: Combine Like Terms

In the given equation, identify and combine like terms. The equation is: \[1 - 3y + y = 5\]The like terms \(-3y\) and \(+y\) can be combined:\[-3y + y = -2y\]Replace these terms in the equation: \[1 - 2y = 5\]

Step 2: Isolate the Variable Term

To isolate the term involving \(y\), subtract 1 from both sides of the equation:\[1 - 2y - 1 = 5 - 1\]This simplifies to:\[-2y = 4\]

Step 3: Solve for the Variable

Now, solve for \(y\) by dividing both sides of the equation by \(-2\):\[-2y = 4\]\[y = \frac{4}{-2}\]This results in:\[y = -2\]

Step 4: Check the Solution

Substitute \(y = -2\) back into the original equation to verify it satisfies the equation:\[1 - 3(-2) + (-2) = 5\]This simplifies to:\[1 + 6 - 2 = 5\]Further simplifying:\[7 - 2 = 5\]And indeed:\[5 = 5\]Since both sides of the equation are equal, the solution \(y = -2\) is verified.

Key Concepts

Combining Like Terms

When you're solving linear equations, one of the first steps often involves combining like terms. Like terms are terms in an equation that have the same variable raised to the same power. In the equation \[1 - 3y + y = 5\]"\(-3y\)" and "\(+y\)" are like terms because they both contain the variable \(y\). To combine, simply add or subtract their coefficients (the numbers in front of the variables). In this case, \(-3y + y\) results in \(-2y\). This simplification transforms the equation into \[1 - 2y = 5\]. Combining like terms is a crucial step because it simplifies the equation, making it easier to solve in subsequent steps.

Isolating Variables

Once you've combined like terms, the next step in solving a linear equation is to isolate the variable. This means getting the term with the variable by itself on one side of the equation. In our example, we need to isolate \(-2y\) in the equation \[1 - 2y = 5\]. Subtract 1 from both sides to accomplish this:\[1 - 2y - 1 = 5 - 1\]. This simplifies to \[-2y = 4\]. By isolating the variable term, you position yourself to solve for the actual variable value next. This step is similar to peeling away layers to focus on the core part of the equation you need to solve.

Checking Solutions

After finding a solution in equation solving, it is vital to check that your answer is correct. You do this by substituting the solution back into the original equation to see if it holds true. For the equation \[1 - 3y + y = 5\], we found \(y = -2\). We substitute \(-2\) back:\[1 - 3(-2) + (-2) = 5\]. Simplifying inside the parentheses first, you get \[1 + 6 - 2 = 5\],which further simplifies to \[7 - 2 = 5\]. Finally, arriving at \[5 = 5\] confirms the solution is indeed correct.Checking your solution helps to ensure accuracy and builds confidence in your problem-solving skills by verifying that the solution satisfies the original equation.