Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$-2 y-5=7$$

Step-by-Step Solution

Answer

The solution to the equation \(-2 y - 5 = 7\) using both the addition and multiplication properties of equality is \(y = -6\).

Step 1: Use Addition Property of Equality

The aim is to isolate the variable term \(-2y\). For that, add \(5\) to both sides of the equation \(-2 y - 5 = 7\). This gives \(-2y = 12\). This is addition property of equality which states that adding identical values to both sides of an equation gives an equivalent equation.

Step 2: Utilize Multiplication Property of Equality

Now, to isolate \(y\), divide both sides of the equation by \(-2\). This gives \(y = -6\) . This is multiplication property of equality, which states that multiplying the same value to both sides of an equation provides an equivalent equation.

Step 3: Check Proposed Solution

Replace \(y\) with \(-6\) in the original equation to check the solution. While doing so, we obtain \(-2 * -6 - 5 = 7\), which simplifies to \(12 - 5 = 7\), which in final simplification gives \(7 = 7\). So, the solution \(y = -6\) is accurate.

Key Concepts

Addition Property of Equality

When solving equations, one of the key properties we rely on is the addition property of equality. This property lets us keep equations balanced by adding the same amount to both sides.
For example, suppose you have the equation \(-2 y - 5 = 7\). We want to solve for \(y\), but first, we need to isolate the \(-2y\) term.
By adding \(5\) to both sides of the equation, we get:

• \(-2y - 5 + 5 = 7 + 5\)
• which simplifies to \(-2y = 12\)

This step ensures that the equation remains true. The addition property of equality is powerful because it helps simplify equations, allowing us to focus on the variable we want to solve. Remember, any operation you perform on one side must be performed on the other.

Multiplication Property of Equality

Once you've used the addition property to simplify the equation, you may need to further isolate the variable using the multiplication property of equality. This concept allows us to multiply or divide both sides of the equation by the same number. As long as you're not multiplying or dividing by zero, the equation stays true.
Continuing with our example, after applying the addition property, we ended up with \(-2y = 12\). To solve for \(y\), we need to get rid of the coefficient \(-2\) by dividing both sides by \(-2\):

• \(-2y \div -2 = 12 \div -2\)
• simplifying to \(y = -6\)

With this final step, we've isolated \(y\), and we can claim \(y = -6\) as our solution. The multiplication property of equality is crucial, especially when dealing with equations that have coefficients.

Checking Solutions

After solving an equation, it's important to verify that the solution is correct. Checking your solution ensures that no mistakes were made during the process and that your answer truly satisfies the original equation.
To verify, substitute the solution back into the original equation. If both sides of the equation are equal, you've found the correct solution.
Using our solution \(y = -6\) in the original equation \(-2y - 5 = 7\):

• Replace \(y\) with \(-6\), resulting in \(-2(-6) - 5 = 7\).
• This simplifies to \(12 - 5 = 7\),which is indeed \(7 = 7\).

Both sides of the equation are equal, signifying that our solution \(y = -6\) is correct. Checking solutions not only confirms our results but also builds confidence in the equation-solving process.