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Chapter 2: Problem 14

For exercises 1-72, (a) solve. (b) check. $$ 9-x=14 $$

Short Answer

Expert verified
The solution is \(x = -5\). Checking the solution shows that both sides of the equation are equal.

Step by step solution

01

Isolate the variable (x)

Start by isolating the variable on one side of the equation. To do this, subtract 9 from both sides of the equation.\[9 - x - 9 = 14 - 9\]This simplifies to:\[-x = 5\]
02

Solve for x

To solve for \(x\), multiply both sides of the equation by -1 to get rid of the negative sign.\[-x \cdot (-1) = 5 \cdot (-1)\]This simplifies to:\[x = -5\]
03

Check the solution

To check if \(x = -5\) is the correct solution, substitute \(x\) back into the original equation and see if both sides are equal.\[9 - (-5) = 14\]This simplifies to:\[9 + 5 = 14\]which is true since 14 equals 14.

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Key Concepts

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

Isolate the variable
When solving a linear equation like the one in our exercise, the first step is to isolate the variable. The variable could be represented by a letter, commonly 'x'. Isolating the variable means getting it alone on one side of the equation. In the exercise, we start with: \( 9 - x = 14 \)To isolate 'x', we need to eliminate the 9 on the left side. We do this by subtracting 9 from both sides:\[ 9 - x - 9 = 14 - 9 \]This simplifies to:\[ -x = 5 \]Here, '-x' is now isolated but still has a negative sign. The goal is to continue until 'x' is fully isolated and positive. Let's see how this is done in the next step.
Solve for x
Once we've isolated the variable with a negative sign, the next step is to solve for 'x'. This often involves performing additional operations to simplify the equation. From our previous step, we have:\[ -x = 5 \]To get rid of the negative sign, we multiply both sides of the equation by -1:\[ -x \times (-1) = 5 \times (-1) \]This simplifies to:\[ x = -5 \]Now, we have successfully solved for 'x'. The value of 'x' that satisfies the equation is -5. In the final step, let鈥檚 confirm that this solution is correct.
Check the solution
Checking the solution is a crucial step to ensure the variable we've solved for is correct. To check, substitute the value of 'x' back into the original equation and see if both sides are equal. Using our solution 'x = -5', we substitute it back in:\[ 9 - (-5) = 14 \]Simplifying that gives us:\[ 9 + 5 = 14 \]Since both sides equal 14, our solution is verified as correct. Performing these steps not only confirms our answer but also reinforces the method of solving linear equations, making the process clearer and easier for future problems.

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Most popular questions from this chapter

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(-2 x+12>36\) Incorrect Answer: \(-2 x+12>36\) $$ \begin{aligned} -12 &-12 \\ \hline-2 x+0 &>24 \\ \frac{-2 x}{-2} &>\frac{24}{-2} \\ x &>-12 \end{aligned} $$

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve \(8 x-2 y=54\) for \(x\). Incorrect Answer: \(8 x\) $$ \begin{array}{r} 8 x-2 y=54 \\ +2 y+2 y \\ 8 x+0=56 \\ \frac{8 x}{8}=\frac{56}{8} \\ x=7 \end{array} $$

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For exercises 53-62, (a) clear the fractions or decimals and solve. (b) check the direction of the inequality sign. $$ 0.3 z+9<15.6 $$
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