Problem 72 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. \(-x-6=-x-1 ;-10\)

Step-by-Step Solution

Answer

No, -10 is not a solution of the equation.

Step 1: Substitute the Given Number

First, we need to substitute the given number \(-10\) into the equation \(-x-6 = -x-1\) by replacing \(x\) with \(-10\). This becomes:\[-(-10)-6 = -(-10)-1\]

Step 2: Simplify Both Sides

Now, simplify both sides of the equation. Start with the left side: \[-(-10)\] becomes \[10\];Then \[10-6 = 4\].Simplify the right side: \[-(-10)\] becomes \[10\];Then \[10-1 = 9\].

Step 3: Compare Both Sides

After simplifying both sides, we compare them:Left side: \(4\)Right side: \(9\)Since \(4eq 9\), the two sides are not equal.

Key Concepts

Substituting Values

The practice of substituting values involves replacing a variable in an equation with a specific number. This allows us to find out the equation's outcome with that particular value. For example, if we have the equation \(-x-6=-x-1\)and we want to check if \(-10\)is a solution, we substitute \(x\) with \(-10\).

• The expression becomes: \[-(-10)-6 = -(-10)-1\].
• By substituting, we can directly see the effects a certain value has on an equation.

Make sure to substitute accurately. Misplacing signs or numbers can lead to incorrect results. Always double-check your substitutions for accuracy.

Simplifying Expressions

Simplifying expressions is crucial as it makes equations easier to analyze and compare. It involves performing basic arithmetic operations to reduce each expression to its simplest form. In our example,

• Start with the substitution result: \[-(-10)-6 = -(-10)-1\].
• First, change \(-(-10)\) to \(10\)because negating a negative number results in a positive one.
• Now, simplify further: \[10-6 = 4\]and \[10-1 = 9\].

Simplifying helps in seeing the equation's structure more clearly. It's often helpful to work step-by-step to avoid errors. Remember that careful simplification is key to comparing sides effectively later on.

Comparing Sides of an Equation

After simplifying, the next step is to compare the results on both sides of the equation. This step determines if the equation is balanced, meaning that both sides have the same value when simplification is complete.

• In our case, we have the left side as \(4\)and the right side as \(9\).
• Since \(4\)and \(9\)are clearly not equal, the initial substituted value does not solve the equation.

Whenever comparing, always ensure both sides are compared after complete simplification. If both simplified results are equal, the number is a solution to the equation. If not, it isn't a solution. Keeping an eye on equality helps in making these determinations accurately.