Problem 20 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$-6 x=-42$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 7\).

Step 1: Identify the Equation

The given equation is \(-6x = -42\). The goal is to isolate \(x\) on one side of the equation.

Step 2: Apply the Multiplication Property of Equality

To isolate \(x\), divide both sides of the equation by \(-6\). This step uses the multiplication property of equality where dividing by \(-6\) is equivalent to multiplying both sides by \(-\frac{1}{6}\).

Step 3: Calculate the Left Side

Divide \(-6x\) by \(-6\): \(\frac{-6x}{-6} = x\)The \(-6\)s cancel each other out, leaving \(x\).

Step 4: Calculate the Right Side

Divide \(-42\) by \(-6\):\(\frac{-42}{-6} = 7\)Since both numbers are negative, their division results in a positive value.

Step 5: Deduce the Solution

After applying the operations, we find that:\(x = 7\)

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics that involves finding the value of the unknown variable that makes the equation true. The process generally involves performing a series of mathematical operations to isolate the variable on one side of the equation. For most equations, the goal is simple: get the variable by itself. This is done through a series of inverse operations, like addition and subtraction, multiplication and division. These operations keep the equation balanced. When solving equations:

• Identify the equation and understand what is being asked.
• Look for operations that are currently affecting the variable.
• Use inverse operations to reverse those effects.

The key is to remember that whatever operation you do to one side, you must also do to the other side to keep things equal. This systematic approach allows you to gradually simplify and solve the equation accurately, revealing the unknown value of the variable.

Isolation of Variable

Isolation of variable refers to the algebraic process of getting the variable alone on one side of the equation to find its value. In our example with \(-6x = -42\), the goal was to solve for \(x\) by getting it by itself. This involves reversing the operations applied to the variable in a step-by-step manner. Here's how it works:

• First, identify any operations attached to the variable. In this case, \(x\) is being multiplied by \(-6\).
• To isolate \(x\), use the inverse operation, which is division in this case. Divide each side of the equation by \(-6\).
• Upon division, the \(-6\) cancels out on the left side of the equation, effectively isolating \(x\) to solve it as \(x \leftarrow 7\).

By following these steps, isolation becomes a straightforward process. It helps you understand the internal logic of math problems and its application in solving equations.

Prealgebra Concepts

Prealgebra introduces essential mathematical concepts that set the foundation for higher levels of algebra. One of these core ideas is the multiplication property of equality, which suggests that you can multiply or divide both sides of an equation by the same nonzero number without changing the equation's solutions. This principle is crucial because it allows for the simplification of equations and the isolation of variables.In practical terms, when you see an equation like \(-6x = -42\), you can use division (which is the inverse of multiplication) to both solve for \(x\) and demonstrate this prealgebra concept. Here’s how: Divide both sides of the equation by \(-6\) to maintain equality and simplify the expression on each side, such that:

• The left side becomes \(x\).
• The right side simplifies to 7.

Understanding and applying the multiplication property of equality gives you a solid groundwork for tackling more complex problems in algebra and beyond, reinforcing problem-solving logic and methodologies used in mathematics.