Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$6 y=2 y-12$$

Step-by-Step Solution

Answer

The solution to the equation \(6y=2y-12\) is \(y=-3\).

Step 1: Isolate variable

Add -2y to both sides of the equation to isolate y terms on one side. This gives us \(6y - 2y = 2y - 2y -12\), which simplifies to \(4y = -12\).

Step 2: Solve for y

To solve for \(y\), divide both sides of the equation by 4. Hence, \(y = -12 /4 = -3\).

Step 3: Check the solution

The final step is to substitute the obtained solution back into the original equation to verify it. If both sides of the equation equal each other when \(y = -3\), the solution is correct. Substitute \(y = -3\) into the original equation: \(6*(-3) = 2*(-3) -12 \), which simplifies to \(-18 = -6 -12\), which simplifies further to: \(-18 = -18\). Therefore, the solution is correct.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is an important concept when it comes to solving linear equations. This property states that you can add the same number to both sides of an equation, and the equation will still hold true. It helps maintain the balance of the equation. For instance, in the original equation \(6y = 2y - 12\), we used this property by adding \(-2y\) to both sides.
This simplifies the equation to \(6y - 2y = 2y - 2y -12\), resulting in \(4y = -12\).

• This property is crucial for moving terms across the equation, especially when the equation needs terms to be rearranged.
• Always ensure that the same value is added to both sides; otherwise, the equality becomes untrue.

It simplifies the problem-solving process by allowing us to eliminate terms on one side of the equation. This sets the stage for more straightforward manipulation, like isolating the variable of interest.

Multiplication Property of Equality

The Multiplication Property of Equality is another essential tool for solving equations, especially when dealing with linear equations. This property indicates that you can multiply or divide both sides of an equation by the same non-zero number, and the equation will remain balanced.
After using the Addition Property of Equality to get \(4y = -12\), we apply the Multiplication Property by dividing both sides by 4 to isolate \(y\).

• This gives us \(y = \frac{-12}{4}\) which simplifies to \(y = -3\).
• Ensure that you never multiply or divide by zero, as this would invalidate the equation.

Utilizing this property is essential when coefficients are present next to the variable, as it allows us to neatly solve for the variable without disrupting the equation's equality.

Variable Isolation

Variable isolation is a fundamental step when solving equations since it allows us to find the specific value of the unknown variable. The goal of variable isolation is to get the variable of interest by itself on one side of the equation.
In the equation \(6y = 2y - 12\), we aimed to isolate \(y\). By subtracting \(2y\) from both sides, we progressively move all the terms involving \(y\) to one side.

• The result is \(4y = -12\), which prepares the equation for solving using division.
• Variable isolation often involves using both the addition and multiplication properties to rearrange and eliminate terms on either side.

This process not only reduces complexity but also makes it easier to verify the solution after solving. Seeing a single variable in the equation is a sign we're close to the solution.

Solution Verification

Solution verification is the final, yet crucial part of solving equations. It involves substituting the obtained solution back into the original equation to verify its correctness.
Once we determined that \(y = -3\), we substituted \(y = -3\) back into the original equation: \(6(-3) = 2(-3) - 12\).

• This simplifies to \(-18 = -6 - 12\), which further simplifies to \(-18 = -18\).
• Since both sides of the equation match, the solution is verified as correct.

This step is essential because it confirms accuracy and prevents errors from going unnoticed.
Solution verification reassures us that the methods and properties used during the solving process were applied correctly, ensuring a reliable and valid solution.