Problem 84 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$5(y-1)+6=-9$$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Coefficient

Start by distributing the 5 across the terms inside the parentheses. This gives: \[ 5(y-1) = 5y - 5 \]So, the equation becomes:\[ 5y - 5 + 6 = -9 \]

Step 2: Combine Like Terms

Combine the constant terms on the left side of the equation. \[ -5 + 6 = 1 \]So, the equation simplifies to:\[ 5y + 1 = -9 \]

Step 3: Isolate the Variable Term

To isolate the term with \(y\), subtract 1 from both sides of the equation:\[ 5y + 1 - 1 = -9 - 1 \]This simplifies to:\[ 5y = -10 \]

Step 4: Solve for the Variable

Divide both sides of the equation by 5 to solve for \(y\):\[ y = \frac{-10}{5} \]This simplifies to:\[ y = -2 \]

Key Concepts

Distributive Property

The distributive property is a crucial tool in algebra that helps simplify expressions and equations. It allows you to distribute a multiplication across terms inside parentheses. In the original problem, we encountered the expression

• \( 5(y-1) + 6 = -9 \).

Using the distributive property, we distribute the \(5\) to both \(y\) and \(-1\). This gives us:

• \( 5 \times y = 5y \)
• \( 5 \times -1 = -5 \)

So, the expression \( 5(y-1)\) becomes \( 5y - 5 \).
This step is critical because it eliminates the parentheses and sets the stage for further simplification.
It is important to remember that the distributive property applies for any number of terms inside the parentheses too!
This can be remembered as \(a(b + c) = ab + ac\) for all numbers \(a\), \(b\), and \(c\).

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by adding or subtracting terms that have the same variable parts. In the given problem, after using the distributive property, we arrived at the expression:

• \( 5y - 5 + 6 = -9 \)

Here, \(-5\) and \(6\) are constants, which means they can be combined.
This becomes:

• \(-5 + 6 = 1 \)

Thus, the equation simplifies to \( 5y + 1 = -9 \).
It’s vital to remember to only combine terms with identical variable parts.
For instance, \(2x\) and \(3x\) are like terms because both have the variable \(x\). However, \(3x\) and \(2y\) cannot be combined because they have different variable parts.
By practicing combining like terms, you'll streamline solving equations and better manage complex expressions.

Isolating Variables

Isolating the variable is a key step in solving linear equations. It involves manipulating the equation so that the variable we're solving for (such as \(y\)) stands alone on one side. After distributing and combining like terms, our equation becomes:

• \( 5y + 1 = -9 \)

To isolate \(y\), you need to perform operations that reverse the equation's components.
In this case, subtract 1 from both sides to get:

• \( 5y = -10 \)

The side that originally had the variable now only has \(5y\), without the added constant.
Next, divide both sides of the equation by the coefficient of \(y\), which is 5:

• \( y = \frac{-10}{5} \)

And finally, simplify:

• \( y = -2 \)

Isolating variables helps in finding exact values in equations, enabling you to solve problems accurately.
This method can be applied universally to solve multi-step equations in algebra.