Problem 58 - Prealgebra

Question

Solve each equation. Check your solution. $$-13=y+5$$

Step-by-Step Solution

Answer

The solution is \(y = -18\).

Step 1: Isolate the variable

To solve the equation \(-13 = y + 5\), we need to isolate \(y\). We do this by subtracting 5 from both sides of the equation. This gives us:\[-13 - 5 = y + 5 - 5\] \[-18 = y\].

Step 2: Verify the solution

Substitute \(y = -18\) back into the original equation to verify the solution.The original equation is:\(-13 = y + 5\).Replace \(y\) with \(-18\):\(-13 = -18 + 5\).Calculate the right side: \(-18 + 5 = -13\).Both sides of the equation are equal, so \(y = -18\) is correct.

Key Concepts

Isolation of Variables

When solving linear equations, isolation of variables is key. Our goal is to get the variable of interest, in this case, \(y\), by itself on one side of the equation. This makes it easier to identify its value.

Here is how you can effectively isolate a variable:

• Perform the opposite operation: If a number is added to your variable, subtract it from both sides. If it's subtracted, add it to both sides.
• Apply this principle until the variable is by itself on one side of the equation.
• Remember to do the chosen operation on both sides of the equation to maintain balance.

In our example, \(-13 = y + 5\), the goal was to isolate \(y\). We noticed that 5 was added to \(y\). To counter this, we subtracted 5 from both sides and got \(-18 = y\). Now, \(y\) is cleanly isolated, and we can see its value.

Checking Solutions

Once you find a potential solution, like \(y = -18\), it's essential to verify that it actually satisfies the original equation. Checking solutions ensures you have the correct answer and not an error from computation.

Verification involves substituting your solution back into the original equation and confirming that both sides are equal:

• Substitute the value back into the equation where the variable was.
• Simplify the equation and verify that both sides are equal.

For \(-13 = y + 5\), substituting \(y = -18\) gives us \(-13 = -18 + 5\). Simplifying the right side results in \(-13\), matching the left side and confirming \(y = -18\) is indeed correct.

Substitution Method

The substitution method is a great way to confirm that your solution is accurate. This process involves replacing the variable in the equation with the value you've determined.

Here’s how you utilize it:

• Take the solved value, in this case, \(-18\), and plug it into your equation in place of the variable.
• After substituting, simplify the equation to check if the left side equals the right side.
• If both sides are equal, the solution is verified.

It's a straightforward process, and it's invaluable for ensuring that you've solved the equation correctly without jumping to conclusions. By substituting \(-18\) for \(y\) in our example, we found that both sides were equal, verifying our solution.

Basic Arithmetic Operations

Understanding basic arithmetic operations such as addition, subtraction, multiplication, and division is vital when working with linear equations. These operations not only help in isolating variables but also in simplifying and verifying solutions.

In equations, follow these basic operations:

• Make use of the order of operations - PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Apply the correct arithmetic operation to maintain the equation's balance.
• Solving often involves reversing operations, like using subtraction to undo addition.

Our example used subtraction to remove 5 from \(y + 5\), leading to the simplification \(-18 = y\). This showcases the importance of correctly applying basic arithmetic operations to both solve and verify the equation.