Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$x+9=16$$

Short answer

x = 7

Step-by-step solution

Isolate the variable x

Subtract 9 from both sides of the equation to isolate the variable x on one side of the equation.

Solve for x

After subtracting 9 from both sides, the equation becomes x = 16 - 9.

Simplify the equation

Simplify the right side of the equation by subtracting the numbers to find the value of x.

Check the solution

Substitute the value of x back into the original equation to verify that the left-hand side equals the right-hand side.

Key concepts

Algebraic Equations

When dealing with algebraic equations, we're working with mathematical statements that show the equality of two expressions. They usually include a variable, which in this case is represented by 'x', and constants or numbers. In the example provided, \(x+9=16\), we have a straightforward linear equation with one variable.

To solve such equations, we aim to find the value of 'x' that makes the equation true. This means finding the number that, when substituting 'x', results in both sides of the equation being equal. Algebraic equations are fundamental as they present basic relationships that can be applied to more complex problems in mathematics and science.

Isolating Variables

The key step in solving an algebraic equation is isolating the variable. This process involves moving all instances of the variable to one side of the equation, and everything else to the opposite side. We achieve this by performing operations that maintain the equation's balance, such as adding, subtracting, multiplying, or dividing both sides by the same number.

In the example \(x+9=16\), we isolate 'x' by subtracting 9 from both sides. It's crucial to apply these operations to both sides so that the equality remains true. Isolation of variables is a technique that is used universally through all forms of algebra.

Simplifying Expressions

After isolating the variable, the next step is often simplifying expressions. Simplification makes the equation easier to understand and solve by combining like terms and reducing expressions to their simplest form.

In the equation \(x+9=16\) and after isolating 'x', we have \(x=16-9\). Simplifying the right side requires us to perform the subtraction, leading to \(x=7\). Simplification can involve arithmetic operations, factoring, expanding, and other algebraic manipulations. The goal is always to reduce complexity while maintaining the integrity of the expression.

Solution Checking

The last step, solution checking, is absolutely essential to ensure that our solution is correct. To check our solution, we substitute our found value of the variable back into the original equation and see if it satisfies the equation.

For our example, we take the solution \(x=7\) and plug it into the original equation to get \(7+9=16\). Since this calculation is accurate, our solution is correct. Solution checking is a method that verifies our work and confirms that we have correctly applied the principles of algebra to find a valid solution for the equation.