Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$8 y+4=2 y-5$$

Step-by-Step Solution

Answer

The solution to the equation \(8y + 4 = 2y - 5\) is \(y = -1.5\).

Step 1: Use Addition Property of Equality

Subtract \(2y\) from both sides of the equation to get \(6y + 4 = -5\).

Step 2: Use Addition Property of Equality again

Now, subtract \(4\) from both sides of the equation to get \(6y = -9\).

Step 3: Use Multiplication Property of Equality

Divide both sides of the equation by \(6\) to solve for y. \(y = -9 / 6 = -1.5\).

Step 4: Check the Solution

To verify the solution, substitute \(-1.5\) for \(y\) in the original equation and see if it makes a true statement. LHS: \(8(-1.5) + 4= -8\). RHS: \(2(-1.5)-5= -8\). Since LHS is equal to RHS, the solution is correct.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a foundational concept for solving linear equations. This property states that when you add or subtract the same number from both sides of an equation, the equality is maintained. Think of it like a balance scale, where each side has to stay equal to keep balance.
In our exercise, we first subtracted \(2y\) from both sides, transforming \(8y + 4 = 2y - 5\) into \(6y + 4 = -5\).
Removing \(2y\) from both sides helped to isolate terms involving \(y\) on one side of the equation, which is a crucial step in solving for the variable.
We then further applied the Addition Property by subtracting \(4\) from both sides to reach \(6y = -9\).
This move was pivotal because it got rid of the constant term on one side, inching us closer to isolating \(y\). Remember, every time you use this property, the goal is to simplify the equation so that you can clearly see the value of the unknown variable.

Multiplication Property of Equality

The Multiplication Property of Equality is another essential tool for solving equations. This property tells us that if we multiply or divide both sides of an equation by the same non-zero number, the mathematical statement remains correct.
In our exercise, after using the Addition Property, we had the equation: \(6y = -9\).
To solve for \(y\), the next logical step was to isolate \(y\) by dividing both sides by \(6\) — the coefficient of \(y\).
This step is vital because it transforms the equation into \(y = -1.5\), revealing the value of the variable directly.
The warning here is to always ensure you're not dividing by zero, as that would disrupt the integrity of the equation. Using the Multiplication Property efficiently will consistently guide you to the value of the unknown in a simple equation.

Solving Equations

Solving equations is a systematic approach to finding the value of unknown variables that make the equation true. Each equation comes with its challenges, but by following step-by-step procedures, you can simplify the process effectively.
In our example, we deciphered the equation \(8y + 4 = 2y - 5\) with the combined use of the Addition and Multiplication Properties of Equality.

• Start by simplifying and rearranging the equation. Reduce it to simpler terms by eliminating variables from one side and constants from the other using the Addition Property.
• Next, use the Multiplication Property to resolve the coefficient of the variable, effectively isolating and solving for the variable.
• Finally, verify your solution by substituting the value back into the original equation to ensure both sides are equal. In our example, substituting \(-1.5\) for \(y\) into both sides of the initial equation confirmed the solution was correct.

Remember, practice makes perfect. With enough exposure to different kinds of equations, solving them becomes second nature. This methodical approach will help increase your efficiency and confidence when tackling equations.