Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ x+8=2(x-2)-x $$

Step-by-Step Solution

Answer

The equation \(x+8=2(x-2)-x\) has no solution.

Step 1: Simplify the Equation

First, simplify the equation by performing the multiplication on the right side. This turns \(2(x-2)-x\) into \(2x-4-x\). Therefore, the equation is now \(x+8=2x-4-x\).

Step 2: Combine Like Terms

Next, combine like terms on both sides of the equation. On the right side of the equation, \(2x-x\) becomes \(x\); on the left side, there are no like terms, so it remains \(x+8\). The equation is now \(x+8=x-4\).

Step 3: Isolate x

In order to solve for \(x\), it needs to be isolated on one side of the equation. Subtract \(x\) from both sides, resulting in \(8=-4\). Check if the equation is true.

Step 4: Check the Solution

By simplifying the equation, it is determined that \(8=-4\). However, this is not possible since \(8\) does not equal \(-4\). Therefore, the original equation has no solution.

Key Concepts

Algebraic Manipulation

When solving linear equations, the objective is to isolate the variable on one side to determine its value. Algebraic manipulation is the process of using mathematical operations to rearrange and simplify an equation. To do this effectively, one must understand the properties of equality and operations, such as addition, subtraction, multiplication, and division.

In our exercise, algebraic manipulation involves performing multiplication first, as per the order of operations, and then moving the terms containing the variable to one side. An important technique in algebraic manipulation is performing the same operation on both sides of the equation to maintain the balance or equality. For instance, if you subtract a number from one side, you must subtract the same number from the other side.

Combining Like Terms

Terms that have the same variable raised to the same power are called like terms. Combining like terms refers to the process of simplifying algebraic expressions by adding or subtracting these terms. It's crucial in solving equations because it helps to reduce the complexity of the expression.

For example, in our given equation, on the right side, we have 2x and -x, which are like terms with the variable x. By combining them (2x - x), we get x. Simplifying the equation by combining like terms makes it easier to isolate the variable. Remember to look out for both the coefficients (the numbers in front of the variables) and the variables themselves when combining like terms—they must match for the terms to be combined.

Checking Solutions

After simplifying and solving an equation, it is important to check the solution to ensure it's correct. This involves substituting the value of the variable back into the original equation to see if it satisfies the equation. If after substituting, both sides of the equation are equal, the solution is correct. If they are not, you either made a mistake or the equation has no solution.

In the exercise, the equation simplifies to 8 = -4, which is a contradiction since 8 can never equal -4. This means there is no value of x that will satisfy the original equation, hence the equation has no solution. Checking the solution helps you to verify your work and confirms the validity of your results.

No Solution Equations

Some equations may have no solution. These are also known as contradiction equations because they present a statement that is always false, no matter what value the variable takes. In a no solution equation, algebraic manipulation will lead to a nonsensical statement, like the one we found in our example: 8 = -4.

Identifying no solution equations is an essential skill, as it prevents unnecessary further calculations. In some cases, you might find an identity, such as 0 = 0, which means the equation has infinitely many solutions, not to be confused with no solutions. Algebraic understanding helps distinguish between identities, no solutions, and equations with a unique solution.