Question

Solve the equation. $$x+17=9$$

Short answer

The solution to the equation \(x+17=9\) is \(x=-8\).

Step-by-step solution

Identify the Eqation

The given equation is \(x+17=9\)

Isolate x

Subtract 17 from both sides of the equation to isolate x. This results in \(x=9-17\)

Simplify the Equation

Solve for x by simplifying the right side of the equation. \(x= -8\)

Key concepts

Algebraic Equations

Algebraic equations are fundamental to solving problems in mathematics. They are formed by equating two algebraic expressions and typically involve variables like \(x\), constants, and arithmetic operations. An example of a simple linear algebraic equation is \(x + 17 = 9\).

In this equation, \(x\) represents an unknown value we are trying to find, 17 is a known constant that is added to our unknown value, and 9 is the constant our equation is equal to. Our goal is to solve for the variable \(x\) to find its value.

Solving these equations requires methodical approaches that often involve several steps. Each step moves us closer to isolating the variable and finding its value, transforming the equation from a question into a clear statement of equality.

Isolation of Variables

In algebra, the isolation of variables is a critical step in solving equations. When we talk about 'isolating' a variable, we mean manipulating the equation in such a way that the variable we're seeking to solve for ends up alone on one side of the equation.

The purpose of this approach is clarity—it converts a complex equation into a simple, readable format where the solution is evident. Looking at the equation \(x + 17 = 9\), our primary aim is to get \(x\) on one side all by itself. To do this, we perform operations that 'undo' what is being done to \(x\). Since 17 is added to \(x\), we subtract 17 from both sides, tipping the balance to make sure the equation remains equal and thus isolating \(x\). This results in an updated equation: \(x = 9 - 17\).

Equation Simplification

Once we've isolated the variable, the next step is equation simplification. This involves performing arithmetic operations to reduce the equation to its simplest form.

Simplifying equations makes the solution clear and easy to understand. After isolating the variable in our example equation to get \(x = 9 - 17\), we now need to simplify by performing the subtraction. Our brains do this automatically with small numbers, but as equations become more complex, it's important to methodically simplify step by step.

In our case, when we subtract 17 from 9, we get \(-8\), giving us the simplest form: \(x = -8\). This is our solution, indicating that the value of the variable \(x\) that satisfies the original equation is -8. By following these steps of isolation and simplification, we can solve algebraic equations effectively.