Problem 53 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$ y-1=5-2 y$$

Step-by-Step Solution

Answer

The solution is \(y = 2\). This solution checks.

Step 1: Add \(2y\) to both sides of equation

First, let's add \(2y\) to both sides of the equation to move all the \(y\) terms to one side. This gives us \(y + 2y = 5 + 1\).

Step 2: Simplify both sides of the equation

Simplify left side by adding 'y' terms together and right side by adding the constants together. This gives us \(3y = 6\).

Step 3: Divide both sides of the equation by 3

Divide both sides by 3 to isolate 'y'. Therefore, the solution is \(y = 6/3\).

Step 4: Simplify 'y'

Simplify 'y', so \(y = 2\). This is the solution.

Step 5: Check the proposed solution

Substitute \(y = 2\) back into the original equation. It should be balanced. \(2 - 1 = 5 - 2*2\); \(1 = 1\). This check is valid, so the solution is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a powerful tool that helps in solving equations. It allows you to add the same quantity to both sides of an equation without changing the equality of the equation. This is crucial when you want to move terms from one side of the equation to the other.

In the original exercise, the equation was given as \( y - 1 = 5 - 2y \). To bring all the terms involving \( y \) to one side, we used the addition property of equality by adding \( 2y \) to both sides. This step transforms the equation into \( y + 2y = 5 + 1 \).

• This eliminates the \(-2y\) term from the right side.
• It helps in grouping like terms together.

Once the terms are gathered appropriately on one side, the problem becomes much simpler.

Multiplication Property of Equality

The multiplication property of equality states that both sides of an equation can be multiplied or divided by the same non-zero number without affecting the balance of the equation. This property helps isolate the variable in order to find its value.

After simplifying using the addition property, the equation becomes \( 3y = 6 \). To solve for \( y \), we utilize the multiplication property of equality by dividing both sides by 3. This means we're multiplying by the reciprocal of 3, or simply dividing the equation by 3.

• This isolates \( y \) on one side, making it \( y = 6/3 \).
• Simplifying gives \( y = 2 \).

The multiplication property of equality is fundamental in getting a clean, precise answer when solving equations.

Checking Solutions

Once you've arrived at a solution, it's crucial to verify that it's correct. This process, known as checking solutions, ensures that the solution satisfies the original equation.

To check the solution from our example, substitute \( y = 2 \) back into the original equation \( y - 1 = 5 - 2y \).

• Substitute \( y = 2 \) into the left side: \( 2 - 1 = 1 \).
• Substitute \( y = 2 \) into the right side: \( 5 - 2\times2 = 1 \).

Since both sides of the equation equate to 1, the solution is verified as correct. Checking solutions not only confirms accuracy but also builds confidence in the techniques used to solve equations.