Problem 70 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. \(5-x=4 x+5\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Rewriting the equation

Firstly, rewrite the given equation in a slightly rearranged form: \(5 - 5 = 4x + x\). The term \(5 - 5\) on the left hand side simplifies to zero, so the equation simplifies to \(0 = 4x + x\).

Step 2: Combining like terms

The next step is to combine like terms on the right hand side of the equation. In this case, \(4x + x\) simplifies to \(5x\), so the equation becomes \(0 = 5x\).

Step 3: Solving for x

To solve for \(x\), divide both sides of the equation by 5. This gives the solution \(x = 0\).

Key Concepts

No Solution Equations

When it comes to linear equations, sometimes you will encounter cases where there are no solutions. This means that no value for the variable will satisfy the equation. Such equations are known as 'no solution equations'.
This happens when simplifying or solving the equation results in a statement that is blatantly false, like saying 3 = 5. If you ever find yourself with an equation where both sides have been simplified to different constants, it indicates that no solution can exist.
It's important to recognize these situations early to avoid unnecessary extra work. Remember, no number will make both sides of a no solution equation equal, hence the term 'no solution'.

True for All Real Numbers

There are special equations that are true for all real numbers, meaning any value of the variable will make the equation true. These are called 'identity equations'.
They arise when you simplify an equation and arrive at a true statement about numbers, such as 4 = 4. This means that the original equation was balanced no matter what the variable's value is.
In simpler terms, if you see the left-hand side and right-hand side reducing to the same expression, it shows that the equation holds true for any real number you plug into the variable. Identifying such equations saves time and effort in solving.

Combining Like Terms

Combining like terms is a fundamental step in simplifying equations and solving for the variable. Like terms are terms that contain the same variable to the same power. For instance, in the equation we had here, both 4x and x are like terms.
You can add or subtract these terms to combine them, which makes the equation simpler. It’s as simple as performing arithmetic operations on coefficients while keeping the variable part unchanged.
For example, combining 4x and x gives you 5x, because the terms have the same variable (x) and you just need to add their coefficients (4 + 1) to get 5.

Rearranging Equations

Rearranging equations is a crucial skill to master in solving algebraic expressions. It involves changing the order of terms to simplify solving for a variable or to make combining terms easier.
The goal is usually to isolate the variable on one side of the equation, which helps to make the solution apparent. In our given exercise, rearranging aimed to separate terms involving the variable from constant terms.
Often, arranging terms efficiently makes it easier to identify operations like combining like terms or dividing to solve for the variable. This helps to streamline the problem-solving process and eliminate potential errors along the way.