Problem 28 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$-q+1=0$$

Step-by-Step Solution

Answer

The solution to the equation is \(q=1\).

Step 1: Rearrange the Equation

The equation can be rearranged to get \(q\) alone on one side. Add \(q\) to both sides of the equation to obtain: \(1=q\).

Step 2: Check Solution

By substituting the calculated \(q\) into the original equation we have: \(-1+1=0\), which holds true.

Key Concepts

Solving Linear Equations

Linear equations are mathematical statements that relate a constant and a variable using basic arithmetic operations. Solving a linear equation involves finding the value of the variable that makes the equation true.
In the given exercise, the equation is \(-q + 1 = 0\). To find the solution, you must isolate the variable (in this case, \(q\)) on one side of the equation. Here's how you can do this:

• Add \(q\) to both sides. This gives: \(1=q\). This step removes the negative sign from the variable and transfers it to the constant side.

Now, the equation clearly shows that \(q = 1\). The variable \(q\) has successfully been isolated, providing the solution for the equation. This simple process reveals how linear equations can be rearranged to isolate variables.

Checking Solution

Once we find a potential solution to a linear equation, it's important to verify that it indeed satisfies the original equation. This builds confidence in our answer and ensures we've solved the problem correctly.
For the equation \(-q + 1 = 0\), we found that \(q = 1\). Checking the solution involves substituting this value back into the original equation:

• Replace \(q\) with 1: \(-1 + 1 = 0\). This simplifies to \(0 = 0\), which actually holds true.

Since the left-hand side equals the right-hand side of the equation, our solution is verified as accurate. Always perform this step after solving equations to catch any possible mistakes.

Basic Algebraic Manipulation

Basic algebraic manipulation is a fundamental skill in mathematics that involves the rearrangement and simplification of equations. These operations make solving for unknown variables possible.
In our example, handling \(-q + 1 = 0\), required simple yet essential manipulation:

• Effectively moving the term containing \(q\) to one side of the equation by adding \(q\) to both sides.
• Ultimately, this rearrangement and simplification yielded the expression \(q = 1\).

Such techniques, like adding, subtracting, multiplying, or dividing terms, adjust the equation while maintaining equality. Practice these basic manipulations to tackle more complex problems with ease.