Problem 114 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 3-x=4 $$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Isolate the Variable

Begin with the equation: \(3 - x = 4\). Our goal is to isolate \(x\) on one side. To achieve this, subtract 3 from both sides of the equation:\[3 - x - 3 = 4 - 3\]This simplifies to:\[-x = 1\]

Step 2: Remove the Negative Sign

Now, we need to solve for \(x\), which currently has a negative sign. Multiply both sides of the equation by \(-1\) to eliminate the negative sign:\[-1(-x) = -1(1)\]This results in:\[x = -1\]

Key Concepts

Isolating a variable

Isolating a variable means getting that variable by itself on one side of the equation. In our example, we start with the equation \(3 - x = 4\). Our goal is to find the value of \(x\). To start, we need to ensure \(x\) is the only term on its side.

• Subtraction helps us remove the number 3 from the left side. To do this, subtract 3 from both sides of the equation: \(3 - x - 3 = 4 - 3\).
• This simplifies the equation to \(-x = 1\).

Isolating variables is an important skill in algebra because it is the first step in solving most equations. By removing any other numbers or terms on the side of the equation with your variable, you make it easier to see what your variable equals.

Solving for x

Solving for \(x\) means finding the value that makes the equation true. After isolating the \(-x\) term in our simplified equation, \(-x = 1\), we must determine what \(x\) equals. At this step, you have the variable appropriately prepared. Now, notice that \(x\) has a negative attached.

• To solve for \(x\), simply eliminate the negative sign by multiplying both sides by \(-1\). This changes all signs in the equation.
• When you multiply, you get \(-1 \times (-x) = -1 \times 1\), which is equivalent to \(x = -1\).

Solving for \(x\) in this method requires understanding the need to balance the equality. Whatever operation you apply to one side of the equation, you must apply to the other, ensuring the equation stays fair and equal.

Negative coefficients

Negative coefficients can often seem tricky, but they are manageable with the right approach. In our equation, \(-x = 1\), the \(-1\) in front of the \(x\) is the negative coefficient. Often, the aim is to transform equations to make the variable's coefficient positive for easier interpretation.
When dealing with negative coefficients:

• Remember that multiplying or dividing both sides of the equation by a negative number reverses the sign.
• In our scenario, multiplying both sides by \(-1\) transforms \(-x = 1\) into \(x = -1\). This switching of signs flips the negative to positive, allowing us to clearly see what \(x\) equals.

Understanding negative coefficients is key in algebra. They can be present in many equations, and knowing how to manage them ensures you can manipulate and solve equations effectively.