Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 9-x=13 $$

Short answer

The solution to the equation \( 9 - x = 13 \) is \( x = -4 \) .

Step-by-step solution

Transpose

Transpose 9 from the left side of the equation to the right. So it becomes, \( x = 9 - 13 \) .

Perform Addition/Subtraction

Subtract 13 from 9. So it becomes, \( x = -4 \) .

Check the solution

To check the solution, substitute \( x = -4 \) into the original equation \( 9 - x = 13 \) . This results in \( 9 - (-4) = 13 \) , which simplifies to \( 13 = 13 \) . The solution checks out.

Key concepts

Transposing Terms

Transposing terms in an equation is a critical skill that is used to isolate the variable of interest on one side of the equation. In the context of solving linear equations like \( 9-x=13 \), it simply means moving a term from one side of the equation to the other. This is done to group like terms together and prepare the equation for simpler operations such as addition or subtraction. When you transpose a term, remember to reverse its sign. For example, in the step-by-step solution, transposing the term '9' from the left side of the equation to the right changes the equation to \( x = 13 - 9 \).

This is a crucial starting point when solving linear equations, as it sets up the equation in a form that we can easily simplify to find the value of the variable.

Addition and Subtraction in Equations

Once we have the equation set up with the variable on one side (thanks to transposing terms), we can use addition and subtraction to solve for the variable. In simple linear equations, these operations help to simplify the equation further to find the solution. The aim here is to isolate the variable and get it on its own.

For instance, after transposing, we get \( x = 13 - 9 \). We then subtract 9 from 13, which gives us \( x = 4 \). This is the step where we simplify the equation to its most basic form—where we have the variable equal to a number. Remember, the order of operations matters here, so follow the mathematical rules correctly to arrive at the right solution.

Solution Checking

Checking your solution is the final and a very important step in solving equations. It's a great practice to ensure that the answer you've obtained is correct. To do this, you plug the value of the variable that you've solved back into the original equation. If both sides of the equation match after substitution, then your solution is correct.

In our exercise, we check our solution \( x = 4 \) by substituting it back into the original equation \( 9 - x = 13 \). This gives us \( 9 - 4 = 13 - 4 \), which simplifies to \( 5 = 5 \). Since both sides of the equation equal each other, our solution is verified. This not only confirms that we have the correct answer, but it also reinforces our understanding of the equation and the steps we took to solve it.