Problem 2 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$w+4=-10$$

Step-by-Step Solution

Answer

The solution to the equation is \( w = -14 \).

Step 1: Identify the Equation

The given equation is \( w + 4 = -10 \). We need to solve for \( w \).

Step 2: Isolate the Variable

Subtract 4 from both sides of the equation to isolate \( w \). This gives us:\[ w + 4 - 4 = -10 - 4 \]Simplifying both sides, we get:\[ w = -14\]

Step 3: Verify the Solution

Substitute \( w = -14 \) back into the original equation to check the solution:\[(-14) + 4 = -10\]The left-hand side simplifies to \(-10\), which matches the right-hand side of the equation, confirming that \( w = -14 \) is correct.

Step 4: Graph on a Number Line

To graph the solution on a number line, place a point or marker at \(-14\) to represent the solution. The number line does not need any shading or additional marks as \( w = -14 \) is a specific value.

Key Concepts

Solving Linear Equations

When we talk about solving linear equations, our goal is to find the value of the unknown variable that makes the equation true. Consider the equation \( w + 4 = -10 \). Here, "\( w \)" is a variable that represents an unknown number. To solve this type of equation, we perform operations that help us isolate the variable on one side of the equation. This process often involves using basic arithmetic operations like addition, subtraction, multiplication, or division.

• Start by identifying the operation performed on the variable. If you have "\( w + 4 \)", it indicates the variable \( w \) is being added to 4.
• To solve for \( w \), perform the opposite operation to isolate it. In this case, subtract 4 from both sides of the equation.
• The result \( w = -14 \) gives us the solution, verifying that substituting \( w \) back into the original equation maintains equality.

Solving equations effectively helps in daily problem-solving, allowing us to determine unknown values by forming correct mathematical relationships.

Number Line Graph

Graphing equations on a number line is a visual representation of solutions. For the equation \( w = -14 \), we can graph this by marking the corresponding position on the number line. Here's how you can do this:

• Draw a horizontal line and mark it with evenly spaced numbers, known as the number line.
• Identify where \(-14\) lies on the number line and place a clear dot or marker at this position. This notation signifies the position where the variable \( w \) equates to its solution value.
• There is no need to shade or indicate ranges because \( w = -14 \) is a single, exact solution rather than a range of solutions.

A number line graph can provide clarity, especially when dealing with equations spanning a range of numbers. It showcases visually where the solution fits within the numerical sequence, making abstract concepts easier to grasp.

Variable Isolation

One of the fundamental steps in solving linear equations is variable isolation. To achieve this, you need to perform operations that leave the variable on one side of the equation by itself. This is crucial for determining the exact value of the variable, as seen in the equation \( w + 4 = -10 \).

• To isolate \( w \), understand that the operation paired with it is addition (adding 4). Perform the inverse operation by subtracting 4 from both sides of the equation. This process is crucial as it maintains the equality of the equation while simplifying it.
• Once the operation is applied, the equation reduces to \( w = -14 \), revealing the isolated variable's value.

Mastering variable isolation techniques ensures solving any linear equation becomes systematic and error-free. This is not only essential in academic problems but also practical life situations where unknowns need to be determined accurately.