Problem 55 - Prealgebra

Question

Solve each equation. Check your solution. $$14=-2 n$$

Step-by-Step Solution

Answer

The solution to the equation is \(n = -7\). Verified by substituting back into the equation: \(14 = 14\).

Step 1: Move the coefficient

The equation given is \(14 = -2n\). To isolate \(n\), we need to get rid of the coefficient \(-2\). We do this by dividing both sides of the equation by \(-2\). This gives us \(-\frac{14}{2} = n\).

Step 2: Simplify the equation

Now, simplify the division: \(-\frac{14}{2} = -7\). Hence, \(n = -7\). This is your solution to the equation.

Step 3: Verify the solution

To ensure our solution is correct, substitute \(n = -7\) back into the original equation. Substitute \(-7\) for \(n\): \(14 = -2(-7)\). The right-hand side results in: \(-2 \times -7 = 14\). Since both sides of the equation are equal, \(n = -7\) is the correct solution.

Key Concepts

Linear Equations

Linear equations are mathematical statements that depict a linear relationship between variables. In its simplest form, a linear equation is expressed as \(ax + b = c\), where \(x\) is the variable, and \(a\), \(b\), and \(c\) are constants. These equations are termed 'linear' because their graph forms a straight line on the Cartesian plane.

There are some key characteristics of linear equations:

• They contain no exponents higher than one, which ensures the graph is a straight line.
• They often involve a single variable, but can also extend to multiple variables, like \(ax + by = c\).
• Solving a linear equation involves finding the value of the variable that makes the equation true.

Understanding linear equations is pivotal in algebra, as they form the basis for more complex mathematical concepts. The example equation, \(14 = -2n\), fits this format, where \(n\) is the variable to be isolated.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. When dealing with linear equations, the goal is to isolate the variable on one side of the equation. Here's a general approach:

• Identify operations: Look for operations affecting the variable like addition, subtraction, multiplication, or division.
• Reverse operations: Use inverse operations to isolate the variable. If the variable is multiplied by a number, divide both sides by that number.
• Simplify: After each operation, simplify both sides of the equation to keep it manageable.

Applying these steps to our example, \(14 = -2n\), we divide both sides by \(-2\) to isolate \(n\), resulting in \(n = -7\). Solving equations requires patience and practice, but it is a fundamental skill for progressing in mathematics.

Checking Solutions

After solving an equation, verifying the solution is an essential step. Checking solutions ensures that the isolated variable indeed satisfies the original equation. Here’s how it works:

• Substitute the solution: Replace the variable in the original equation with the solution obtained.
• Perform the arithmetic: Calculate both sides of the equation to ensure they are equal.
• Confirm equality: If both sides match, the solution is correct. If not, revisit your calculations.

For example, in \(14 = -2n\), substituting \(n = -7\) gives \(14 = -2(-7)\), which simplifies to \(14 = 14\). Both sides are equal, confirming \(n = -7\) as a valid solution. Checking solutions builds confidence in your answer and provides a valuable opportunity to catch errors.