Problem 59 - Prealgebra

Question

Solve each equation. Check your solution. $$18=20+x$$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Isolate x

Subtract 20 from both sides of the equation to solve for x. Start with the given equation: \(18 = 20 + x\). By subtracting 20 from both sides, we have:\[ 18 - 20 = x \] which simplifies to \(x = -2\).

Step 2: Check the Solution

Substitute \(x = -2\) back into the original equation to verify the solution is correct. Start with the original equation:\(18 = 20 + x\). Substitute \(x = -2\) to get: \[ 18 = 20 + (-2) \] Simplifying the right-hand side gives \(18 = 18\), which confirms that the solution is correct.

Key Concepts

Steps to Solve Equations

Solving equations can feel daunting at first, but it becomes much easier when broken down into steps. Here are the basic steps to solve equations:

• First, identify your goal: isolate the variable, which is often represented by a letter such as \( x \).
• Next, perform operations to both sides of the equation to isolate the variable. This often involves adding, subtracting, multiplying, or dividing.
• Continue simplifying the equation until the variable is on one side and the solution is a number on the other side.

For example, consider the equation given: \(18 = 20 + x\). To solve for \( x \), we want \( x \) by itself on one side. You see that subtracting 20 from both sides helps achieve this, leading you to find that \( x = -2 \). The key is performing each operation equally on both sides to maintain the equation's balance.

Checking Solutions

Once you have found a solution to an equation, it's important to check your work. This ensures that mistakes haven't been made during the process. Checking solutions typically involves substituting your answer back into the original equation. In the case of our equation \(18 = 20 + x\), after finding \( x = -2 \), you plug \( -2 \) back into the equation like this:\[18 = 20 + (-2) \]Simplifying the right side should give you the original left side of the equation, confirming that both sides are equal, in this case, \(18 = 18\). When both sides are equal, your solution is correct.

Linear Equations

Linear equations are a fundamental part of algebraic studies and are important in understanding math as a whole. A linear equation is one in which the variable is raised to the power of one, making it a straight line when graphed.Some properties of linear equations include:

• They have no exponents on the variable other than one.
• They can usually be written in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.
• Solving them often involves straightforward manipulation to solve for the variable.

Our example \(18 = 20 + x\) is a simple linear equation. Understanding this concept helps in not only solving problems but predicting how equations behave.

Basic Algebra

Basic algebra forms the foundation of all higher-level mathematics. It involves using letters and symbols to represent numbers and express relationships between them. This element of abstraction may seem challenging at first, but it's powerful for solving real-world problems. Key components of basic algebra include:

• Understanding variables and how they can change.
• Using operations like addition, subtraction, multiplication, and division.
• Applying the distributive, associative, and commutative properties.

Being comfortable with these ideas will make solving every algebra problem—including equations like \(18 = 20 + x\)—easier and more intuitive. Mastery of basic algebra allows for a smoother progression to more advanced topics.