Question

Solve the equation if possible. $$ 6 s-11=-2 s+5 $$

Short answer

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

Step-by-step solution

Bring like terms together

Start by bringing like terms together. The variable \(s\) is on both sides of the equation. To bring the \(s\)-terms together, you should add \(2s\) to both sides of equation. This will give you: \[ 6s + 2s = 5 + 11 \]

Simplify the equation

Simplify both sides of the equation. On the left side, \(6s + 2s\) equals \(8s\). On the right side, \(5 + 11\) equals \(16\). This simplifies the equation to: \[ 8s = 16 \]

Solve for \(s\)

Finally, you will solve for \(s\) by dividing both sides of the equation by 8. This gives you:\[ s = 2 \]

Key concepts

Understanding Algebraic Expressions

At the heart of algebra lies the understanding of algebraic expressions. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like s in our exercise), and operators (such as +, -, *, and /). These expressions describe relationships and can represent numbers in terms of variables. For example, in the equation 6s - 11 = -2s + 5, 6s and -2s are algebraic expressions representing multiples of the variable s.

The power of algebraic expressions becomes clear when we use them to express real-life scenarios, such as calculating distances at different speeds, or figuring out the total cost if we know the cost per item. Learning to manipulate these expressions effectively enables us to solve for unknown values, ultimately revealing the answers to the problems at hand.

Simplifying Equations

The process of equation simplification is crucial to solving problems efficiently. In mathematics, simplification means reducing complexity while maintaining equivalence. When you simplify an equation, you're seeking to make the expression on each side of the equals sign as straightforward as possible. This often involves combining like terms and eliminating unnecessary parts.

In our problem, simplification comes into play when we gather like terms and add them together, turning 6s + 2s into 8s and 5 + 11 into 16. These simplified forms are much easier to work with. Remember, to simplify correctly, it's essential to understand the properties of equality and the correct use of operations. This careful attention to the process leads to accurate solutions.

Isolating the Variable

When solving equations, the goal is to isolate the variable – to get the unknown by itself on one side of the equation. Doing so reveals its value. In our exercise, we isolate the variable s by performing operations that 'undo' what has been done to s. Since s is multiplied by 8, we perform the opposite operation: we divide both sides of the equation by 8.

This balancing act is based on the properties of equality, which state that what is done to one side of an equation must be done to the other to maintain equality. By isolating the variable, we have easily found that s equals 2, solving the equation. Understanding how to isolate variables is a fundamental skill in algebra that you will use time and time again to unlock the solutions to various problems.