Problem 57 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ x+1=5 $$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Understand the Equation

The equation given is \(x + 1 = 5\). This is a simple linear equation where \(x\) is an unknown quantity and the goal is to find its value.

Step 2: Isolate the Variable

To solve for \(x\), you need to get \(x\) by itself on one side of the equation. To do this, subtract 1 from both sides of the equation.

Step 3: Perform the Operation

Subtract 1 from both sides of the equation: \[ x + 1 - 1 = 5 - 1 \] This simplifies to \(x = 4\).

Step 4: Verify the Solution

Substitute \(x = 4\) back into the original equation to ensure the solution is correct. The original equation is \(x + 1 = 5\). Substitute the solution: \[ 4 + 1 = 5 \] Since the left side equals the right side, the solution is verified.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical sentences that express a relationship between variables without any exponents or powers. They usually involve terms connected by addition or subtraction symbols. In this case, the equation is \(x + 1 = 5\). Here, "\(x\)" is the variable; the unknown value that we want to find.

To distinguish a linear equation, look for these characteristics:

• The highest power of the variable is always 1.
• There is a constant term, which in this equation is 5.
• It forms a straight line when graphed on a coordinate plane.

Linear equations often model real-world scenarios such as calculating distance, budget, or converting units.

Isolation of Variables in Linear Equations

The term "isolation of variables" refers to the process of solving for one variable in an equation. For linear equations, this involves getting the unknown variable alone on one side of the equation.

In our example, \(x + 1 = 5\), the goal is to isolate \(x\). This is achieved by performing inverse operations:

• Identify the constant added or subtracted from \(x\). Here, it's 1 added to \(x\).
• Apply the opposite operation to both sides of the equation. Subtract 1 from both sides: \(x + 1 - 1 = 5 - 1\).

With that simple step, we've isolated \(x\) and found that \(x = 4\). This method can be used for more complex linear equations as well, always focusing on reversing the operations to isolate the variable.

Verification of Solutions for Linear Equations

Once a solution is found, it is crucial to verify it. Verification ensures that the solution satisfies the original equation, confirming its correctness.

To verify, simply substitute the solution back into the original equation. For \(x = 4\) in \(x + 1 = 5\):

• Replace \(x\) with 4 in the equation: \(4 + 1 = 5\).
• Check if both sides of the equation are equal: 5 equals 5.

Since both sides match, this confirms that our solution \(x = 4\) is correct. Verification not only checks accuracy but also boosts confidence in solving equations.