Problem 39 - Prealgebra

Question

Graph the solution of each equation on a number line. $$\frac{y}{-4}=-1$$

Step-by-Step Solution

Answer

The solution is \( y = 4 \); plot it as a point on the number line at 4.

Step 1: Solve for y

To isolate \( y \), we multiply both sides of the equation by \(-4\): \[ \frac{y}{-4} = -1 \rightarrow y = -1 \times (-4). \] Therefore, \( y = 4 \).

Step 2: Draw a Number Line

Sketch a horizontal line, marking evenly spaced increments at least from 2 to 6 to include the solution point 4.

Step 3: Plot the Solution

Identify the point \( y = 4 \) on the number line you drew. Clearly mark this point with a dot and label it \( y = 4 \).

Key Concepts

Number Line

In the context of graphing equations, a number line is a simple yet powerful tool used to visually demonstrate the values of variables, such as those found in solutions to equations. When plotting solutions, the number line offers a clear and intuitive way to represent numbers.

To draw a number line:

• First, sketch a horizontal line.
• Select an appropriate range that will capture your solution. It's always a good idea to extend slightly beyond the solution for a clearer view.
• Divide the line into equal increments, marking integers and pertinent fractions as needed.

For our given exercise, because the solution obtained was 4, you would mark numbers around 4 to represent this solution efficiently. Place a dot directly on the line at the number 4, which signifies the exact solution to your equation.

Solving Linear Equations

Linear equations involve variables raised to the power of one, making their graphs straightforward lines when graphed in two dimensions. Decoding a simple linear equation like \( \frac{y}{-4} = -1 \) involves isolating the variable to understand what specific value of 'y' can satisfy the equation.

The key steps include:

• Identifying the variable that needs to be isolated—in this case, 'y'.
• Maneuvering the equation using basic algebraic operations such as addition, subtraction, multiplication, or division.
• Strictly following the rule that any operation done on one side of the equation must also be done to the opposite side, ensuring equivalent transformations.

In this particular equation, multiplying both sides by \(-4\) effectively cancels out the division by \(-4\) on the left side, thus solving for 'y', yielding \(y = 4\). This step-by-step approach illustrates the logical flow and preserves the equality.

Algebraic Solutions

Algebraic solutions involve systematically finding the value of an unknown variable that makes an equation true. It's about unraveling what the equation is hiding. Often, you'll employ algebraic properties that cancel or simplify certain terms or coefficients.

Techniques in obtaining an algebraic solution involve:

• Understanding and identifying what operation is required to isolate a variable.
• Applying inverse operations—essentially performing the opposite operation—such as using multiplication to cancel a fraction.
• Always checking your solution by plugging it back into the original equation to ensure it satisfies the equation, confirming the accuracy of your solution.

In our problem, the algebraic manipulation required multiplying both sides by \(-4\) allowed us to reveal that \(y = 4\). This highlights how finding solutions is less about memorizing steps and more about understanding the underlying properties that govern algebra.