Problem 25 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$3 a+4=2(a-5)+15$$

Step-by-Step Solution

Answer

The solution to the equation is \( a = 1 \).

Step 1: Simplify Both Sides

Start by simplifying both sides of the equation. On the right side, distribute the 2 to both terms inside the parentheses: \[ 2(a-5) = 2a - 10 \] Now substitute back into the equation:\[ 3a + 4 = 2a - 10 + 15 \] Combine like terms on the right side to simplify further:\[ 3a + 4 = 2a + 5 \]

Step 2: Isolate the Variable Terms

Next, you want to isolate the variable terms on one side of the equation. Subtract \(2a\) from both sides to keep the variables on the left:\[ 3a - 2a + 4 = 2a - 2a + 5 \] This simplifies to:\[ a + 4 = 5 \]

Step 3: Solve for the Variable

Finally, solve for \( a \) by subtracting 4 from both sides:\[ a + 4 - 4 = 5 - 4 \]This simplifies to:\[ a = 1 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step in solving linear equations. It involves combining and reducing terms in an expression to make the equation easier to work with. When you look at an equation, you'll often see terms that are like each other, such as numbers or variables. These like terms can be combined.
For example, in the equation \(3a + 4 = 2a + 5\), notice that \(2a\) and \(15 - 10\) are terms that can be simplified first.
### Steps to Simplify an Expression- Identify like terms on both sides of the equation.- Combine like terms by performing the necessary addition or subtraction.Doing this reduces clutter and makes it easier to isolate the variable in the later steps. Remember that simplifying expressions not only involves addition but can also include subtraction, multiplication, and division when combining terms.
It's important to simplify as much as possible before moving on to solving the equation.

Distributive Property

The distributive property is a fundamental concept that allows you to simplify expressions involving parentheses. This property states that multiplying a sum by a number is the same as multiplying each addend individually and then adding the products.
Mathematically, it can be stated as:\[ a(b + c) = ab + ac \]In the equation provided, the distributive property is used to simplify \(2(a - 5)\). When you distribute the 2, it becomes:\[ 2 \times a + 2 \times (-5) = 2a - 10 \]### Key Points about the Distributive Property- **Allows simplification:** It helps break down complex expressions into simpler components.- **Avoids mistakes:** By distributing first, you minimize the risk of errors later on.- **Applies to subtraction:** Remember that the distributive property works with subtraction as well as addition, as seen in \(2(a - 5)\).Using the distributive property correctly ensures that your expressions are as straightforward as possible, making the overall process of solving equations more manageable.

Isolating Variables

Isolating the variable is an essential step in solving linear equations, as it allows you to find the value of the unknown quantity. The goal is to have the variable on one side of the equation by itself and all the constant terms on the other.
### Steps to Isolate a Variable- **Move variable terms to one side:** In \(3a + 4 = 2a + 5\), moving \(2a\) to the left side by subtraction gives \(3a - 2a\).- **Eliminate constant terms:** Subtract any numbers on the same side as the variable to move constants to the other side. In \(a + 4 = 5\), subtracting 4 from both sides results in \(a = 1\).Once the variable is isolated on one side, you can directly solve for it. It's important to perform the same operation on both sides of the equation to maintain balance and equality. This ensures that the solution you find is correct.
Isolating variables can sometimes involve additional steps like factoring or dividing, but the basic principle remains the same—getting the variable by itself is key to solving equations.