Problem 95
Question
Determine whether (a) \(x=-1\) or (b) \(x=2\) is a solution of the equation. $$x-8=-9$$
Step-by-Step Solution
Verified Answer
The value \(x=-1\) is a solution for the equation \(x-8=-9\), while \(x=2\) is not a solution.
1Step 1: Verify \(x=-1\)
Substitute \(x=-1\) into the equation and see if the left-hand side equals the right-hand side. The given equation is \(x-8=-9\), so substituting \(x=-1\) gives us \(-1-8\), which equals \(-9\). The left-hand side does equal the right-hand side (-9) when \(x=-1\), confirming that \(x=-1\) is a solution for the equation.
2Step 2: Verify \(x=2\)
Next, substitute \(x=2\) into the equation and check whether the left-hand side equals the right-hand side. The original equation is \(x-8=-9\), so substituting \(x=2\) gives \(2-8\), which equals \(-6\). Since \(-6\) does not equal \(-9\), \(x=2\) is not a solution for the equation.
Key Concepts
verify solutionssubstitute valuescompare left and right sides
verify solutions
When solving an equation, it is essential to verify potential solutions. Verifying solutions means checking if a proposed value satisfies the equation. In our example problem, we have to determine whether specific values, like \(x = -1\) and \(x = 2\), are solutions to the equation \(x - 8 = -9\).
This process involves substituting the proposed value into the equation and performing the arithmetic calculation on both sides. If both sides of the equation equal each other after making substitutions, the proposed value is verified as a solution. This step ensures that we are identifying the correct values that satisfy the equation. Let's look at how substituting values plays into this.
This process involves substituting the proposed value into the equation and performing the arithmetic calculation on both sides. If both sides of the equation equal each other after making substitutions, the proposed value is verified as a solution. This step ensures that we are identifying the correct values that satisfy the equation. Let's look at how substituting values plays into this.
substitute values
Substituting values is a crucial method in verifying if they are solutions. With substitution, we replace the variable with a specific number and perform the necessary arithmetic.
Imagine our problem: determine if \(x = -1\) is a solution. We substitute \(-1\) in place of \(x\) in the equation \(x - 8 = -9\). This transforms it into \(-1 - 8 = -9\). Calculate the left side: \(-1 - 8 = -9\), which fits perfectly. Thus, \(x = -1\) is indeed a solution.
Substituting \(x = 2\) follows similarly: substitute \(2\) in the place of \(x\) to get \(2 - 8 = -9\). Compute the left side: \(2 - 8 = -6\), which does not equal the right side \(-9\). Therefore, \(x = 2\) is not a solution. Substitution is straightforward and is key to understanding whether or not a number satisfies an equation.
Imagine our problem: determine if \(x = -1\) is a solution. We substitute \(-1\) in place of \(x\) in the equation \(x - 8 = -9\). This transforms it into \(-1 - 8 = -9\). Calculate the left side: \(-1 - 8 = -9\), which fits perfectly. Thus, \(x = -1\) is indeed a solution.
Substituting \(x = 2\) follows similarly: substitute \(2\) in the place of \(x\) to get \(2 - 8 = -9\). Compute the left side: \(2 - 8 = -6\), which does not equal the right side \(-9\). Therefore, \(x = 2\) is not a solution. Substitution is straightforward and is key to understanding whether or not a number satisfies an equation.
compare left and right sides
After substituting values, the next step is to compare the left and right sides of the equation. This comparison confirms whether the two sides are equal.
For instance, in our problem with the equation \(x - 8 = -9\), after substituting \(x = -1\), we ended with both sides simplifying to \(-9\). They match perfectly, signifying \(x = -1\) is a valid solution.
However, with \(x = 2\), substituting led to the left side \(-6\), which did not match the right side \(-9\). Since they are not the same, \(x = 2\) is not a solution. This diligent comparison helps confirm the correctness of proposed solutions and ensures accuracy in mathematical problem-solving. The importance of this step cannot be overstated as it solidifies one's understanding of not just potential solutions, but the logical reasoning behind them.
For instance, in our problem with the equation \(x - 8 = -9\), after substituting \(x = -1\), we ended with both sides simplifying to \(-9\). They match perfectly, signifying \(x = -1\) is a valid solution.
However, with \(x = 2\), substituting led to the left side \(-6\), which did not match the right side \(-9\). Since they are not the same, \(x = 2\) is not a solution. This diligent comparison helps confirm the correctness of proposed solutions and ensures accuracy in mathematical problem-solving. The importance of this step cannot be overstated as it solidifies one's understanding of not just potential solutions, but the logical reasoning behind them.
Other exercises in this chapter
Problem 95
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