Problem 45 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$3 a+2 a+a=7-13$$

Step-by-Step Solution

Answer

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

Step 1: Combine Like Terms on the Left

The left side of the equation is composed of several terms involving the variable \(a\): \(3a + 2a + a\). To simplify, combine these terms. Adding them together gives \((3+2+1)a = 6a\).

Step 2: Simplify the Right Side

The right side of the equation is \(7 - 13\). Simplify this expression to find the resulting number. Subtracting 13 from 7 yields \(-6\). Thus, the right side simplifies to \(-6\).

Step 3: Set the Simplified Equation

Now that both sides are simplified, set the equation \(6a = -6\). This equation can now be solved for \(a\).

Step 4: Solve for a

To find the value of \(a\), divide both sides of the equation by 6 to isolate \(a\). This gives \(a = \frac{-6}{6}\). Simplify this fraction to find \(a = -1\).

Key Concepts

Combining Like Terms

Combining like terms is a foundational concept in prealgebra that involves merging terms with the same variables and exponents. In the equation given, we see the expression \(3a + 2a + a\) on the left side. Here, "like terms" are those that include the variable \(a\).
To combine these:

• Add the coefficients (numbers in front of the variable \(a\)), which are 3, 2, and 1.
• When combined, these coefficients become \(3 + 2 + 1 = 6\).
• This results in the simplified term \(6a\).

Combining like terms helps reduce the complexity of an equation, making it easier to solve. Remember, you can only combine terms that have identical variable parts, such as \(a^2\) with \(a^2\) or \(b\) with \(b\).

Simplifying Expressions

Simplifying expressions involves reducing an equation or a part of it to its simplest form. In the exercise, both sides of the equation need simplification for easier solving.
The left side, \(3a + 2a + a\), simplifies to \(6a\) by combining like terms. The right side, \(7 - 13\), simplifies through basic arithmetic.

• Subtract 13 from 7.
• This gives the simplified result of \(-6\).

Simplification reduces clutter within equations and highlights the underlying relationships between terms. It's especially useful in equations because it paves the way for easier problem-solving by reducing the overall complexity.

Solving Linear Equations

Solving linear equations is the process of finding the value of the unknown variable that makes the equation true. After simplifying each side of the equation \(6a = -6\), the next step is finding the value of \(a\). Governing principles of algebra tell us to

• Isolate the variable on one side of the equation to find its value.
• In this case, divide both sides by the coefficient of \(a\) (which is 6) to solve for \(a\).

This gives us \(a = \frac{-6}{6}\), which simplifies to \(a = -1\).
Linear equations can be simple or complex but always follow logical steps to find the solution, making this technique valuable in solving countless problems in mathematics.