Problem 83 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$3(y-4)+5=-4$$

Step-by-Step Solution

Answer

The solution is \( y = 1 \).

Step 1: Distribute Constants

Distribute the constant 3 across the terms inside the parentheses. This means multiplying each term inside the parenthesis by 3: \[ 3(y-4) = 3y - 12 \] Thus, the equation becomes: \[ 3y - 12 + 5 = -4 \]

Step 2: Combine Like Terms

Combine the constant terms on the left side of the equation. Simplify \(-12 + 5\) to get \(-7\). The equation now looks like this: \[ 3y - 7 = -4 \]

Step 3: Isolate the Variable Term

Add 7 to both sides of the equation to isolate the term with the variable:\[ 3y - 7 + 7 = -4 + 7 \]This simplifies to: \[ 3y = 3 \]

Step 4: Solve for the Variable

Divide both sides of the equation by 3 to solve for \(y\):\[ \frac{3y}{3} = \frac{3}{3} \]This simplifies to: \[ y = 1 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to remove parentheses by distributing a multiplication over an addition or subtraction inside the parentheses. This is particularly useful when dealing with equations because it helps to simplify expressions.

Imagine you have an expression like \(3(y-4)\). According to the distributive property, you multiply the 3 with each term inside the parenthesis. This means:

• Multiply 3 by \(y\)
• Multiply 3 by -4

So, \(3(y-4)\) becomes \(3y - 12\). By using the distributive property, we convert a complex expression into a simpler algebraic equation without parentheses.

Practicing the distributive property helps build strong algebraic skills, especially when facing longer or more complicated equations. Remember, distributing simplifies your work and makes it easier to solve further steps in the problem.

Combining Like Terms

Once you have distributed any constants, the next logical step in simplifying an equation is combining like terms. Like terms are terms in an equation that have the same variable raised to the same power. In the equation \(3y - 12 + 5 = -4\), after distribution, we have like terms that need to be combined.

Notice the numbers \(-12\) and 5. Both of these terms are constants, meaning they do not have any variables attached to them. To combine these like terms, simply add or subtract the coefficients as you would with any ordinary numbers:

• Combine \(-12 + 5\) to get \(-7\)

Now, our equation is reduced to \(3y - 7 = -4\).

Combining like terms helps clean up equations and makes it easier to isolate variables. This process reduces the complexity of the equation and prepares it for the final steps of solving for the variable. It’s like tidying up your work, so you have a clear path to the solution.

Variables in Equations

Variables are symbols used to represent numbers in equations. In our example, the variable is \(y\). Solving the equation involves isolating this variable, so we can find out what number it represents.

After combining like terms, the equation\(3y - 7 = -4\) has a clear path towards isolating \(y\). Here's how to do it:

• Step 1: Add 7 to both sides to get rid of the constant term on the left: \(3y - 7 + 7 = -4 + 7\).
• This leaves us with \(3y = 3\).
• Step 2: Finally, divide both sides by 3 to solve for \(y\): \(\frac{3y}{3} = \frac{3}{3}\).
• This simplifies to \(y = 1\).

Remember, solving equations is like a balance game. Whatever you do to one side, you must do to the other. This ensures the equation remains equal. Through this process, you systematically reduce the equation until the variable is isolated and you have your solution.