Problem 34 - Prealgebra

Question

Solve each equation. Check your solution. $$17.6=y+11.5$$

Step-by-Step Solution

Answer

The solution is \(y = 6.1\).

Step 1: Identify the Equation

The given equation is \(17.6 = y + 11.5\). Our goal is to isolate the variable \(y\) on one side of the equation.

Step 2: Isolate the Variable

To isolate \(y\), subtract 11.5 from both sides of the equation:\[17.6 - 11.5 = y + 11.5 - 11.5\]Simplifying this, we get:\[6.1 = y\]Thus, \(y\) is equal to 6.1.

Step 3: Check the Solution

Substitute \(y = 6.1\) back into the original equation to verify:\[17.6 = 6.1 + 11.5\]Calculate the sum on the right:\[6.1 + 11.5 = 17.6\]Since both sides of the equation are equal, the solution \(y = 6.1\) is correct.

Key Concepts

Isolation of Variables

When solving equations, one of the most important steps is isolating the variable. The variable is typically represented by a letter, and it is what we are solving for. Here, the variable is denoted as \(y\) in the equation \(17.6 = y + 11.5\). Our goal is to have \(y\) alone on one side of the equation to clearly identify its value.To isolate \(y\), we need to "remove" the 11.5 that is added to it. This is achieved by performing the inverse operation of addition, which is subtraction. By subtracting 11.5 from both sides of the equation, we ensure that we maintain the equation's balance.

• Subtract from both sides: \(17.6 - 11.5 = y + 11.5 - 11.5\)

This results in the simplified equation \(6.1 = y\), effectively isolating the variable \(y\). Once isolated, it becomes easy to see that \(y = 6.1\). This forms the basis of solving most equations, where getting the variable alone is key to finding its value.

Checking Solutions

After finding a solution for the variable, checking your work ensures that the solution is indeed correct. For our example, after calculating \(y = 6.1\), we verify this value by substituting it back into the original equation.Substitute the value of \(y\) into the equation:

• \(17.6 = 6.1 + 11.5\)

Adding \(6.1 + 11.5\) yields 17.6, confirming that both sides of the equation are equal. This confirmation is crucial, especially in more complex equations, as it acts as proof that our solution is accurate. Checking solutions is like double-checking your work; it provides confidence and assurance that you haven't made an error during calculation.

Linear Equations

Linear equations are among the simplest forms of equations, characterized by variables raised only to the power of one. They usually appear in the format \(Ax + B = C\) where \(A\), \(B\), and \(C\) are constants.In our exercise, the linear equation presented was \(17.6 = y + 11.5\). The task was to solve for \(y\).

• These equations can be solved through basic arithmetic, involving addition, subtraction, multiplication, or division.

What makes linear equations straightforward is the consistent approach with which they can be solved: reorganizing terms to get the variable alone on one side.Working with linear equations builds a foundation for understanding more complex algebraic equations. Mastery of these concepts provides essential problem-solving skills useful in various mathematical contexts.