Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$-3 y-7=-1$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = -2\).

Step 1: Apply Addition Property of Equality

The addition property of equality states that when you add the same number to both sides of an equation, the equation remains true. To isolate \(y\), add 7 to both sides of the equation \(-3y - 7 = -1\). That gives you \(-3y - 7 + 7 = -1 + 7\). This simplifies to \(-3y = 6\).

Step 2: Apply Multiplication Property of Equality

The multiplication property of equality states that when you multiply the same number to both sides of an equation, the equation remains true. To completely isolate \(y\), divide both sides of the equation by -3. That gives you \(-3y/-3 = 6/-3\). This simplifies to \(y = -2\).

Step 3: Check the Solution

To verify the proposed solution, substitute -2 for \(y\) in the original equation. This gives you \(-3(-2) - 7\), which simplifies to 6 - 7 equals -1. Thus, the solution \(y = -2\) is correct as it satisfies the original equation.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental concept in algebra that allows us to maintain the balance of an equation while we work to isolate a variable. Essentially, it says that you can add the same amount to both sides of an equation and it will still hold true.

For example, in the equation \( -3y - 7 = -1 \), you can add 7 to both sides to begin isolating the variable \(y\). Here's the process in action:

• Original Equation: \( -3y - 7 = -1 \)
• Add 7 to both sides: \( -3y - 7 + 7 = -1 + 7 \)
• Simplified Equation: \( -3y = 6 \)

By adding 7 to both sides, the \( -7 \) and \(+7 \) on the left side cancel out, allowing us to move one step closer to finding the value of \(y\). Applying this property is crucial for systematically solving for variables and reaching the solution.

Multiplication Property of Equality

Moving on from the addition property, we employ the multiplication property of equality when we need to further isolate the variable. This property tells us that you can multiply (or divide) both sides of an equation by the same nonzero number without changing the equation's solution.

In our ongoing example, divide both sides by -3 to solve for \(y\):

• Equation to Solve: \( -3y = 6 \)
• Divide by -3: \( \frac{-3y}{-3} = \frac{6}{-3} \)
• Isolated Variable: \( y = -2 \)

This step isolates \(y\) on one side of the equation, giving us the solution. Dividing by -3 is necessary here because \(y\) was originally multiplied by -3 in the equation. By applying the inverse operation, we counteract the multiplication and find \(y\)'s value.

Isolating Variables

Isolating the variable, which can also be referenced as 'solving for the variable', is the process of manipulating the equation to get the variable by itself on one side, with a numerical answer on the other. This involves using a combination of addition, subtraction, multiplication, and division to move all other numbers away from the variable.

The ultimate goal is to have the variable in the form of \(x = \text{number}\) or \(y = \text{number}\), which indicates the solution to the equation. Here's how it worked in our sample equation:

• Start with the addition property to eliminate subtraction: \( -3y - 7 + 7 = -1 + 7 \)
• Then apply the multiplication property to cancel out multiplication: \( \frac{-3y}{-3} = \frac{6}{-3} \)

After applying both properties appropriately, we get \(y = -2\), which is the isolated variable.

Checking Solutions in Algebra

Finally, an essential step after isolating the variable and obtaining a solution is to check if that solution is correct. We do this by substituting the solution back into the original equation and confirming that both sides of the equation are equal.

For our equation, we substitute \(y = -2\) and calculate:

• Substitute in Original Equation: \( -3(-2) - 7 \)
• Perform the Operations: \( 6 - 7 \)
• Check If Equal to -1: \( -1 = -1 \)

Since both sides equal -1 after the substitution, our solution \(y = -2\) is verified as correct. Checking solutions is critical in ensuring that no arithmetic errors were made during the solving process and that the solution is indeed valid for the original equation.