Select the correct answer from each drop-down menu. Consider the equation below. [tex]$-2(5 x+8)=14+6 x$[/tex] The equation was solved using the following steps. [tex]
\begin{aligned}
\text { Step 1: } & -10 x-16 & =14+6 x \
\text { Step 2: } & -16 x-16 & =14 \
\text { Step 3: } & -16 x & =30 \
\text { Step 4: } & x & =\frac{30}{-16} \
\text { Step 5: } & x & =-\frac{15}{8}
\end{aligned}
[/tex] Complete the statements below with the process used to achieve steps 1-4. Step 1: [ ] -2 to [tex]$5 x$[/tex] and 8. Step 2: [ ] 6x. Step 3: [ ] 16. Step 4: [ ] -16.
Answer (2)
Step 1: Distribute -2 to both terms inside the parenthesis: − 2 × 5 x = − 10 x and − 2 × 8 = − 16 .
Step 2: Subtract 6 x from both sides of the equation: − 10 x − 6 x = − 16 x .
Step 3: Add 16 to both sides of the equation: 14 + 16 = 30 .
Step 4: Divide both sides of the equation by -16.
The solution is x = − 8 15 .
Explanation
Problem Analysis We are given the equation − 2 ( 5 x + 8 ) = 14 + 6 x and the steps taken to solve for x . Our task is to describe the algebraic operations performed in each step.
Step 1 Explanation Step 1: To get from − 2 ( 5 x + 8 ) to − 10 x − 16 , we distribute the − 2 to both terms inside the parenthesis: − 2 ∗ 5 x = − 10 x and − 2 ∗ 8 = − 16 . So, we multiply -2 to 5 x and 8.
Step 2 Explanation Step 2: To get from − 10 x − 16 = 14 + 6 x to − 16 x − 16 = 14 , we subtract 6 x from both sides of the equation: − 10 x − 6 x = − 16 x . So, we subtract 6x.
Step 3 Explanation Step 3: To get from − 16 x − 16 = 14 to − 16 x = 30 , we add 16 to both sides of the equation: 14 + 16 = 30 . So, we add 16.
Step 4 Explanation Step 4: To get from − 16 x = 30 to x = − 16 30 , we divide both sides of the equation by -16. So, we divide by -16.
Final Answer Therefore: Step 1: Multiply -2 to 5 x and 8. Step 2: Subtract 6x. Step 3: Add 16. Step 4: Divide by -16.
Examples
Understanding how to solve equations step-by-step is crucial in many real-world scenarios. For example, imagine you're trying to determine how many hours you need to work to afford a new gadget. If the gadget costs $150, and you earn $15 per hour but also have to pay $30 in taxes, the equation would be 15 h − 30 = 150 . Solving this equation using the same algebraic steps helps you find the number of hours, h , you need to work. This principle applies to budgeting, financial planning, and many other practical situations.
To solve the equation − 2 ( 5 x + 8 ) = 14 + 6 x , we first distribute − 2 , then isolate the x terms by subtracting 6 x , add 16 , and finally divide by − 16 to find (x = -\frac{15}{8}.
;
