Problem 66 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Solve the equation. (Lesson 3.1) $$ -6+x=-15 $$

Step-by-Step Solution

Answer

The solution is x = -9.

Step 1: Add 6 on both sides of the equation

Add 6 to both sides of the equation to eliminate -6 from the left side. This gives the new equation: -6 + x + 6 = -15 + 6. In other words, x = -15 + 6.

Step 2: Simplify the right side of the equation

Simplify the right side of the equation by doing the arithmetic, -15 + 6 = -9. Hence, the value of x is -9, i.e. x = -9.

Step 3: Check the solution

Substitute x = -9 back into the original equation to check if the equation holds. -6 + (-9) = -15. This equation is correct, confirming that x = -9 is a solution to the original equation.

Key Concepts

Linear Equations

A linear equation is an equation that models a straight line when graphed. Typically, it is in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. They are called linear because they produce a straight line on a graph. In the example \(-6 + x = -15\), is a simple linear equation.Linear equations are fundamental in algebra because they are the most basic type of equation to solve. Understanding how to handle these types of equations is crucial for dealing with more complex equations later on. Solving them often involves finding a number that can replace the variable \( x \) to satisfy the equation.Solving linear equations requires careful step-by-step manipulation, which usually starts with identifying the operations involved on the variable and adjusting the equation to isolate \( x \).

Algebraic Manipulation

Algebraic manipulation involves rearranging equations using arithmetic operations to make them easier to solve. In our example, the goal is to isolate the variable \( x \). This involves moving terms around to get \( x \) by itself on one side of the equation.

• Addition/Subtraction: In our example, we start by adding 6 to both sides. This cancels out the -6 on the left side.
• Simplification: Simplifying both sides of the equation requires performing the arithmetic operations correctly. In this case, adding 6 to -15 gives -9, simplifying the equation to \( x = -9 \).

These processes lead to the solution by logically working through each operation step-by-step. Practicing algebraic manipulation strengthens your ability to solve future equations efficiently.

Inverse Operations

Inverse operations are mathematical operations that 'undo' each other. This concept is central to solving equations. In the context of the example, we want to relieve \( x \) of the term attached to it, which was -6.To achieve balance in the equation and isolate \( x \), we performed the inverse of subtraction, which is addition. Thus, adding 6 to both sides canceled out the -6, effectively undoing that part of the equation. This reveals the power of inverse operations:

• Keep the Equation Balanced: Whatever you do to one side of an equation, do to the other to maintain equality.
• Undo Operations: Using the inverse operation helps to isolate the variable step-by-step until you find its value.

Understanding inverse operations simplifies solving linear equations, allowing you to methodically address different terms influencing the variable.