Problem 32 - Prealgebra

Question

Solve each equation. Check your solution. $$3-y=13$$

Step-by-Step Solution

Answer

The solution is \(y = -10\).

Step 1: Identify the Equation

The given equation is \(3 - y = 13\). Our goal is to solve this equation for \(y\).

Step 2: Isolate the Variable

To isolate \(y\), we need to move the constant term on the left side of the equation to the right side. Subtract 3 from both sides of the equation. \(3 - y - 3 = 13 - 3\). This simplifies to \(-y = 10\).

Step 3: Solve for the Variable

To solve for \(y\), we need to make \(y\) positive. Multiply both sides by -1 to get \(y = -10\).

Step 4: Check the Solution

Substitute \(y = -10\) back into the original equation to check if it holds true: \(3 - (-10) = 3 + 10 = 13\). The left side equals the right side, confirming that the solution is correct.

Key Concepts

Solving Equations

Understanding how to solve equations is a fundamental skill in mathematics. An equation is a statement that asserts the equality of two expressions. To solve an equation means to find the value(s) of the variable that make the equation true. In this process, our main goal is to determine the unknown variable, which in this case is typically represented by letters like "x" or "y." The key is to manipulate the equation to isolate the variable on one side of the equal sign while ensuring the equality holds. This often involves operations like addition, subtraction, multiplication, or division applied to both sides of the equation. The aim is always to simplify the equation step by step while maintaining the balance that the equality represents.

Isolating Variables

A crucial part of solving equations is isolating the variable you want to solve for. This involves rearranging the equation such that the variable stands alone on one side of the equation. When you have an equation like \(3 - y = 13\), you can apply basic arithmetic operations to both sides to simplify. Here's how it works:

• Identify the Constant: In \(3 - y = 13\), the constant on the side with the variable is \(3\). Our goal is to move it to the other side of the equation.
• Move the Constant: Subtract \(3\) from both sides to begin isolating \(y\). This gives us \(-y = 10\).

Once you've isolated the variable, make sure if you have negative signs they are correctly handled. Multiply or divide both sides by \(-1\) if necessary to make the variable positive, as in this example where you end up with \(y = -10\). Isolating variables often requires careful arithmetic to ensure the final variable is in its simplest, positive form.

Checking Solutions

After solving the equation, it's important to check your solution to ensure it is correct. This step verifies that the solution you obtained indeed satisfies the original equation.To do this, substitute the value of the variable back into the original equation and simplify both sides. For this example:Substitute \(y = -10\) back into \(3 - y = 13\). This gives us:\[3 - (-10) = 13 \]Simplifying the left side gives:\[3 + 10 = 13 \] Since both sides of the equation are equal, \(13 = 13\), it ensures our solution \(y = -10\) is indeed correct. Checking solutions not only reinforces the accuracy of your result but also provides confidence in the method used. Always make checking the last step in solving equations to confirm your understanding and calculations are precise.