Problem 21 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$-6+y=-17$$

Step-by-Step Solution

Answer

The solution is \(y = -11\).

Step 1: Identify the Equation

The equation as presented is \(-6+y=-17\). The goal is to isolate \(y\).

Step 2: Apply the Addition Property of Equality

The addition property of equality states that we can add the same amount to both sides of an equation and the equality will still hold. In this case, to get \(y\) alone on one side of the equation, add 6 to both sides of the original equation: \(-6+y+6=-17+6\).

Step 3: Simplify the Equation

Upon adding, the equation simplifies to \(y=-11\).

Step 4: Check the Solution

Substitute \(y=-11\) back into the original equation to verify that it holds true. \(-6+(-11) = -17\). -17 equals -17, so the solution is correct.

Key Concepts

Solving Linear Equations

When we talk about solving linear equations, we refer to finding the value of the unknown variable that makes the equation true. These equations are called 'linear' because they represent a straight line when graphed on a coordinate plane.

The process involves performing operations that maintain the equation's balance, meaning whatever change you make to one side of the equation, you must do the same to the other side.

• First, identify the unknown variable that you need to solve for.
• Then, use mathematical operations such as addition, subtraction, multiplication, or division to rearrange the equation and isolate the variable.
• Keep the equation balanced by performing each operation on both sides.

If done correctly, you'll end up with the variable on one side of the equation and a numerical value on the other, which is the solution. For instance, the equation \( -6+y=-17 \) can be solved for \( y \) by adding 6 to both sides, eventually finding that \( y=-11 \).

Isolate Variables

To isolate the variable means to rearrange the equation so that the variable you are solving for is by itself on one side of the equal sign.

Tips for Isolating Variables

When you're trying to isolate a variable:

• Remember the ultimate goal is to have the variable on one side and the numbers on the other.
• Determine which operations will help move the numbers across the equal sign.
• operate with precision to maintain the balance of the equation.

In our example \( -6+y = -17 \), adding 6 to both sides isolates the variable \( y \) on the left side, resulting in \( y=-11 \). This step is crucial in the process of equation solving because it provides the solution to the variable in question. Remember, isolating the variable should be done with methodical steps to avoid errors.

Equation Checking

Once you believe you have solved the equation, it's essential to perform equation checking to confirm that your solution is correct.

How to Check Your Solution:

To check the solution of an equation:

• Take the value of the variable you've found and substitute it back into the original equation.
• Simplify the equation with the substituted value to see if both sides equal each other.
• If they do, your solution is correct; if not, re-examine your steps to find any possible error.

For example, after solving the equation \( -6+y=-17 \) and getting \( y=-11 \), you substitute \( y \) with -11 in the original equation: \( -6+(-11) \). If both sides simplify to the same number (in this case, -17), the solution \( y=-11 \) is verified as correct. This checking step is crucial to ensure accuracy in your work.