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Chapter 2: Problem 6

Solve each equation, and check your solution. $$ x-9=8 $$

Short Answer

Expert verified
The solution is x = 17, and it checks out.

Step by step solution

01

Isolate the variable

To solve the equation, isolate the variable by adding 9 to both sides of the equation. This will cancel out the -9 on the left side. So, we have: \[ x - 9 + 9 = 8 + 9 \]
02

Simplify both sides

Now simplify both sides of the equation. The left side simplifies to x and the right side simplifies to 17: \[ x = 17 \]
03

Check your solution

To verify the solution, substitute x with 17 in the original equation and check if both sides are equal: \[ 17 - 9 = 8 \] Since 8 equals 8, the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables: The Key to Solving Linear Equations
To isolate x:, you need to undo the operation that involves x. In this case, you have x - 9. To get x alone, add 9 to both sides of the equation: \[ x - 9 + 9 = 8 + 9 \] Now the equation simplifies like this: \[ x = 17\]. This simple step of isolating the variable helps you to clearly identify what x is.
Checking Solutions for Accuracy
Once you find a solution, always check your work. This helps to confirm that the solution is correct. For the equation \: \: x - 9 = 8 \: \: we found that x = 17. To check this, substitute 17 back into the original equation: \[ 17 - 9 = 8 \]. If both sides of the equation are equal, then your solution is correct. In this example, 8 equals 8. Therefore, x = 17 is indeed the correct solution. Checking solutions like this helps you catch any mistakes you might have made in your calculations.
Simplifying Equations to Make Solving Easier
Simplifying equations means making them as straightforward as possible by combining like terms and removing unnecessary elements. In our example, after isolating the variable and simplifying both sides of the equation \: \: x - 9 + 9 = 8 + 9 \: \: we get: \: \ \[ x = 17 \ \]. Simplification helped to reach the solution quickly. Simplifying equations reduces the risk of errors and makes it easier to identify the variable. Always aim for the simplest form, and solving equations will become a much smoother process.

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Most popular questions from this chapter

When the lesser of two consecutive integers is added to three times the greater, the result is 43. Find the integers.

If four times a number is added to \(8,\) the result is three times the number, added to \(5 .\) Find the number.

By what number is it necessary to multiply both sides of each equation to isolate \(x\) on the left side? Do not actually solve. $$ -\frac{8}{3} x=-11 $$

Write each phrase as a mathematical expression using \(x\) as the variable. The quotient of \(-6\) and a nonzero number

Decide whether each of the following is an expression or an equation. If it is an expression, simplify it. If it is an equation, solve it. \(\begin{array}{ll}{\text { (a) } 5 x+8-4 x+7} & {\text { (b) }-6 y+12+7 y-5} \\\ {\text { (c) } 5 x+8-4 x=7} & {\text { (d) }-6 y+12+7 y=-5}\end{array}\)
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