Problem 96 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -4 k-6=7 $$

Step-by-Step Solution

Answer

The solution to the equation is \(k = -\frac{13}{4}\).

Step 1: Add 6 to Both Sides

To isolate the variable terms, add 6 to both sides of the equation: \(-4k - 6 + 6 = 7 + 6\)This simplifies to: \(-4k = 13\).

Step 2: Divide by -4

Now, divide both sides of the equation by -4 to solve for \(k\):\(\frac{-4k}{-4} = \frac{13}{-4}\).This simplifies to: \(k = -\frac{13}{4}\).

Key Concepts

Equation Solving

Equation solving is a fundamental concept in algebra that involves finding the value of an unknown variable that makes the equation true. It requires performing a sequence of mathematical operations on both sides of the equation. These operations, such as addition, subtraction, multiplication, or division, help to simplify the equation and isolate the variable.

Here are some useful tips for solving equations:

• Balance the Equation: Whatever operation you do to one side of the equation, do it to the other side to maintain equality.
• Simplify Each Step: Simplify the equation as much as possible at every step. This often involves combining like terms or clearing fractions.
• Check Your Solution: Substitute the solution back into the original equation to verify that it satisfies the equation.

Equation solving not only helps in algebra but also in many real-world problems where relationships between quantities need to be understood and calculated.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised to the power of one, like the one given: \(-4k - 6 = 7\). These are simple yet powerful equations that represent a straight line when graphed on a coordinate plane. Linear equations are characterized by their straightforward format and can usually be written in the form: \(ax + b = c\).

Understanding the components of a linear equation helps in solving them effectively:

• Coefficients: These are the numbers in front of variables (like \(-4\) in \(-4k\)), indicating how many times the variable is added.
• Constants: These are the standalone numbers either added or subtracted in the equation (like \(-6\) and \(7\)).

Linear equations form the baseline for more complex systems of equations and are an essential building block in algebra. They enable us to model and solve a wide array of problems, from simple everyday questions to complex scientific calculations.

Isolation of Variables

Isolation of variables is a key step in solving equations. It means rearranging the equation so that the variable we are solving for is on one side, isolated by itself. By doing so, we can directly determine the value of the variable. Let's walk through the process using the original problem:

- Start by moving constants to one side: In \(-4k - 6 = 7\), add \(6\) to both sides to counteract the \(-6\). This gives \(-4k = 13\).- Next, remove the coefficient of the variable: Divide both sides by \(-4\) to get \(k = -\frac{13}{4}\).Practicing isolation of variables helps to build the skills necessary to solve more complicated problems in algebra. It is a powerful tool for untangling and solving equations systematically, ensuring you always end up with the correct value for the unknown variable.