Problem 21 - Introductory and Intermediate Algebra for College Students 4th

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 to the equation is \(y = -11\).

Step 1: Identify the Addition Property of Equality

The addition property of equality states that if you add the same amount to both sides of an equation, the equation remains true. In this equation, we can add 6 to both sides in order to isolate the variable \(y\).

Step 2: Apply the Addition Property of Equality

We apply the addition property of equality by adding 6 to both sides of the equation. This gives us the following equation: \(-6 + 6 + y = -17 + 6\). Simplified, this expression is \(y = -11\).

Step 3: Check the Solution

We check the solution by substituting \(y = -11\) back into the original equation, which gives us \(-6 + -11 = -17\). Simplified, this is \(-17 = -17\), which is correct. Hence, our solution is validated.

Key Concepts

Solving Linear Equations

When you're faced with a linear equation like \( -6 + y = -17 \), the goal is to find the value of the variable that makes the equation true. Solving linear equations is a fundamental skill you'll often use in algebra. The process typically involves a series of steps designed to simplify the equation until the variable is isolated on one side. This means that everything that does not contain the variable should be on the opposite side of the equation.

For our given example, you need to get the variable \(y\) by itself on one side of the equation to find its value. The 'addition property of equality' is a tool that helps in doing so. It allows you to add the same number to both sides of the equation without changing the equality. Therefore, you preserve the balance of the equation, much like ensuring both sides of a scale have equal weight.

Isolate the Variable

To isolate the variable in an equation, identify what's being added to or subtracted from the variable and then perform the opposite operation on both sides of the equation. This doesn't just apply to integers but to any other terms, such as fractions or variables.

In our example \( -6 + y = -17 \), the number -6 is being added to \(y\). The opposite operation is addition, so we'll add 6 to both sides, following the principle of maintaining equality. Performing \( -6 + 6 \), we're essentially canceling the -6 on the left, leaving the variable \(y\) isolated on one side: \( y = -11\). Our goal here is to clear the path for \(y\) to stand alone, thereby finding its value.

Equation Checking

Once you've isolated the variable and found a solution, it's crucial to check that your solution is correct. This verification step involves plugging your solution back into the original equation to see if it yields a true statement.

For our exercise, we've determined that \(y = -11\). To check, we substitute \(y\) with -11 in the original equation: \( -6 + (-11) = -17\). Simplify the left side to get \( -17\), which matches the right side of the equation. This confirms our solution, as both sides are equal, showing that -11 is indeed the correct value for \(y\) in the given equation. Remember, a correct solution should always maintain the balance of the equation, just as it did before we started our operations.