Problem 29 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 2. $$ -6-y=-13 $$

Step-by-Step Solution

Answer

The solution is \(y = 7\).

Step 1: Add 6 to both sides

The goal is to isolate the variable \(y\). Start by adding 6 to both sides of the equation to help eliminate the -6 on the left side. This gives us:\[-6 - y + 6 = -13 + 6\] which simplifies to\[-y = -7\].

Step 2: Multiply by -1 to isolate y

To solve for \(y\), we need to eliminate the negative sign in front of \(y\). Multiply both sides of the equation by -1 to get:\[y = 7\].

Step 3: Check the solution

Substitute \(y = 7\) back into the original equation to ensure it is a valid solution:\[-6 - 7 = -13\].The left-hand side simplifies to -13, which matches the right-hand side, confirming that \(y = 7\) is correct.

Key Concepts

Solving Linear Equations

When it comes to solving linear equations, the objective is to find the value of the variable that makes the equation true. Linear equations usually take the form of \(ax + b = c\). Solving them involves performing operations to both sides of the equation to keep it balanced. This ensures the equation's integrity while transforming it into a simpler form.

• Identify the equation and look for the terms involving the variable.
• Use addition, subtraction, multiplication, or division to simplify these terms.
• Work the equation until the variable is isolated on one side with a coefficient of 1.

In our example, we began solving the equation \(-6 - y = -13\) by aiming to isolate \(y\) and to add 6 to eliminate the constant term paired with the variable.

Isolation of Variables

The isolation of variables is a crucial step in solving equations. The goal is to rearrange the equation so that the variable in question stands alone on one side. This process typically involves reversing operations that have been applied to the variable:

• Identify any constants or coefficients associated with the variable.
• Use inverse operations to remove these numbers.

In our example, we first addressed the \(-6\) by adding 6 to both sides, which resulted in the intermediate step: \(-y = -7\). To fully isolate \(y\), it was necessary to eliminate the minus sign by multiplying by \(-1\). Doing this led us to \(y = 7\), successfully isolating \(y\). This demonstrates the power of inverse operations, as they help peel away layers binding the variable.

Checking Solutions

After isolating the variable and determining a potential solution, it is essential to check your work. This is a verification step to ensure the solution satisfies the original equation. You should substitute the solution back into the equation:

• Start by replacing the variable with your found solution.
• Simplify both sides of the equation to see if they match.
• If they equate, your solution is correct; if not, revisit previous steps.

In our example, substituting \(y = 7\) back into the original equation \(-6 - y = -13\) gave us a left-hand side of \(-13\), identical to the right-hand side. This confirmed our solution was accurate, showcasing the importance of this final check.