Problem 80 - Algebra A Combined Function

Question

Match each equation in the first column with its solution in the second column. Items in the second column may be used more than once. a. all real numbers b. no solution C. 0 $$ -x+15=x+15 $$

Step-by-Step Solution

Answer

The solution is 0.

Step 1: Understand the Equation

The equation given is \(-x + 15 = x + 15\). This is a linear equation with one variable, \(x\). Our task is to find the value of \(x\) that makes this equation true.

Step 2: Rearrange the Equation

To solve the equation, we need to move all terms involving \(x\) to one side. Start by adding \(x\) to both sides to get:\(-x + x + 15 = x + x + 15\), which simplifies to \(15 = 2x + 15\).

Step 3: Isolate the Variable

Subtract 15 from both sides to isolate terms with \(x\):\(15 - 15 = 2x + 15 - 15\), resulting in \(0 = 2x\).

Step 4: Solve for the Variable

Divide both sides of the equation by the coefficient of \(x\), which is 2:\(\frac{0}{2} = \frac{2x}{2}\) simplifies to \(0 = x\).

Key Concepts

Equation Solving

In mathematics, linear equations are equations of the first degree, meaning they contain variables raised to the power of one.
Our goal when solving these equations is to find the value of the variable that satisfies the equation. For example, in the equation
\(-x + 15 = x + 15\), we need to determine the value of \(x\) that makes both sides identical.
This process often involves various steps such as moving terms around, isolating the variable, and simplifying the expression.
Understanding each stage ensures that you're correctly applying mathematical rules to find that particular value of the variable.
It's like untangling a riddle where each move brings you closer to the solution – ensuring every step is followed carefully and logically is key to arriving at the correct value.

Variable Isolation

Variable isolation is a central technique in solving linear equations. It involves manipulating the equation such that the variable you're solving for is on one side, and all other terms are on the opposite side of the equation.
In our example, starting with the equation \(-x + 15 = x + 15\), you want to move every term involving \(x\) to one side.
This can be accomplished by adding \(x\) to each side, transforming the equation to \(15 = 2x + 15\).

• By rearranging terms efficiently, you create a simpler equation where you can easily identify the value of the variable.
• Once terms are balanced on either side, other modifications – like subtraction or division – can be employed for further simplification.

Thus, isolating the variable not only simplifies the process but also clearly highlights the path to the solution, allowing you to see the next step.

No Solution Equations

Sometimes, during the course of solving an equation, you may encounter situations where it leads to an impossible or false statement. In such cases, we say the equation has no solution.
These occur when the attempt to simplify results in a contradiction, like \(0 = 5\) - clearly a falsehood.
Equations of this nature do not have any value of \(x\) that could satisfy them.

• This indicates that due to how the equation is originally set up, there's no number that can substitute in and balance both sides.
• Recognizing these early can save time and effort because trying to force a solution will only lead to confusion.

Thus, an equation with no solution means the terms are inherently unsolvable under normal arithmetic.

All Real Numbers Solutions

In contrast to equations with no solutions, some equations resolve in such a way that any real number can be substituted for the variable, and the equation will still hold true.
They are called "all real numbers solutions". Consider if an equation simplifies to a statement like \(15 = 15\); it's always true, no matter what \(x\) is.
This implies every real number is a potential solution.

• Such an outcome usually happens when the variable terms cancel each other out properly and only the constant true statement remains.
• They emphasize the equality of both sides without any dependence on a specific value for the variable.

Understanding this helps in identifying when you've reached a comprehensive solution – realizing that the problem is set up, so the variable can take on any value without altering the equality.