Problem 37 - Prealgebra

Question

Graph the solution of each equation on a number line. $$-42=-7 x$$

Step-by-Step Solution

Answer

The solution is \(x = 6\), marked on a number line at 6.

Step 1: Isolate the Variable

To graph the solution on a number line, we first need to solve the equation for the variable \(x\). Start by isolating \(x\) on one side of the equation. Given the equation: \[-42 = -7x\]we need to get \(x\) by itself. To do this, divide both sides of the equation by \(-7\): \[ x = \frac{-42}{-7} \].

Step 2: Simplify the Expression

Simplify the fraction \(\frac{-42}{-7}\) to find the value of \(x\). When you divide \(-42\) by \(-7\), you get \[ x = 6 \].

Step 3: Graph the Solution on a Number Line

Now that we have \(x = 6\), we can graph this solution on a number line. Draw a horizontal line and place tick marks at equal intervals to represent numbers. Mark a point or a dot above \(6\) to indicate the solution of the equation.

Key Concepts

Number Line

A number line is a visual representation of numbers placed in a straight, horizontal line. It helps us understand the order and relative magnitude of numbers visually. The line extends infinitely in both directions and is marked at equal intervals to represent integers such as 1, 2, 3, and so on.

When we graph a solution, like in our original exercise, we focus on marking specific points on the line. For instance, once we solve the equation and find that the solution for our variable is 6, we use the number line to illustrate where this value sits. This involves drawing a dot directly above the tick mark labeled "6." It is important to remember:

• Always keep the spacing between numbers consistent.
• Use arrows at the ends of the line to show it continues indefinitely.
• Your line can expand further if you need to include larger numbers or decimals.

A number line is a fundamental tool in visualizing and understanding the results of solving equations. It helps in comparing numbers and can also be used in operations such as addition and subtraction.

Isolation of Variables

Solving equations often requires isolating the variable – this means getting the variable by itself on one side of the equation. In our exercise, we started with the equation \[-42 = -7x\].

The first step in isolation is to perform basic arithmetic operations that change the equation without altering its equality. In this case, divide both sides by \(-7\).

• Division is our tool to remove factors multiplied with our variable "x".
• Performing the same operation on both sides is crucial to keep the equation balanced.

After dividing, we simplify the equation to \(x = 6\). This makes it clear and easy to see what value the variable should take.Isolating the variable is key to solving linear equations. It involves reversing operations to work backward from the original equation. This process not only finds a solution but also helps to understand how equations are structured and manipulated.

Graphing Solutions

Once we've isolated the variable and simplified our equation like we did to get \(x = 6\), the next step is graphing this solution. While the solution \(x = 6\) is clear, graphing it adds another layer to understanding.

• Start by drawing a straight line on paper or a digital tool.
• Add tick marks at equal distances to denote integers.
• Find the tick mark labeled "6" and place a prominent dot or point above it.

Graphing on a number line provides a visual confirmation that helps verify your solution is correctly found. It also visually demonstrates where the solution lies in relation to other numbers. This graphic representation is particularly helpful in understanding how altering one part of an equation can shift the solution's position on the number line.Understanding graphing helps solidify knowledge in solving equations and allows you to clearly communicate where solutions exist in the greater numerical landscape.