Problem 12 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$x+5=-3$$

Step-by-Step Solution

Answer

The solution is \(x = -8\).

Step 1: Isolate the Variable

Start with the equation \(x + 5 = -3\). To isolate \(x\), you need to undo the addition of 5. Do this by subtracting 5 on both sides of the equation: \(x + 5 - 5 = -3 - 5\). This simplifies to \(x = -8\).

Step 2: Check the Solution

Substitute \(x = -8\) back into the original equation to verify the solution. Substitute \(-8\) for \(x\) to get: \(-8 + 5 = -3\). Simplify the left side: \(-3 = -3\). Since both sides are equal, the solution \(x = -8\) is correct.

Step 3: Graph on a Number Line

Draw a horizontal line to represent the number line. Locate and mark the point \(-8\) on this line. You could use a dot or a circle to indicate that \(x = -8\) is the solution.

Key Concepts

Checking Solutions

When solving linear equations, it’s crucial to verify your solution to confirm its correctness. After you've found a potential solution, always substitute it back into the original equation. This process helps ensure that the solution satisfies the equation.
For example, if you solved the equation and found that \(x = -8\), substitute \(-8\) back into the equation \(x + 5 = -3\). The left-hand side becomes \(-8 + 5\), which simplifies to \(-3\), matching the right-hand side. Thus, the equation holds true, confirming that your solution is correct.
By double-checking your work, you can catch any errors and improve your problem-solving skills. Consistently checking solutions builds confidence and accuracy in math.

Isolating the Variable

Isolating the variable in an equation is a primary goal to find its solution. To do this, you need to perform operations that reverse those applied to the variable. The objective is to have the variable alone on one side of the equation.
In the equation \(x + 5 = -3\), you see addition of 5. To isolate \(x\), you need to get rid of this 5. Do so by subtracting 5 from both sides, which looks like this: \(x + 5 - 5 = -3 - 5\). This simplifies down to \(x = -8\).
It's important to remember that whatever operation you do to one side of the equation, you must do to the other side as well. Doing so keeps the equation balanced, which is critical when isolating variables.

Graphing on a Number Line

Once you have found the solution to a linear equation, graphing it on a number line can provide a visual representation.
To graph \(x = -8\):

• Draw a horizontal line to act as your number line.
• Include numbers in increments along it for reference, like \(-10\), \(-9\), \(-8\), etc.
• Locate the solution \(-8\) on this line.
• Mark the point \(-8\) with a dot or circle to show it is the solution.

Graphing equations helps reinforce the concept of solutions being specific values within the numeric system. It also provides a link between abstract numbers and their spatial representation, which is helpful for understanding and visualizing solutions.