Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's begin by defining the variables and translating the problem statement into mathematical equations.
1. Defining the Variables:
Let [tex]\( x \)[/tex] represent Peter's age in years.
Let [tex]\( y \)[/tex] represent Phil's age in years.
2. Translating the Statements:
- Phil's age is 7 years more than [tex]\(\frac{1}{5}\)[/tex] of Peter's age.
Mathematically, this can be written as:
[tex]\[ y = \frac{1}{5} x + 7 \][/tex]
- Four times Phil's age is 2 years less than twice Peter's age.
Mathematically, this can be written as:
[tex]\[ 4y = 2x - 2 \][/tex]
3. Substituting the First Equation into the Second:
We have [tex]\( y \)[/tex] expressed in terms of [tex]\( x \)[/tex] from the first equation:
[tex]\[ y = \frac{1}{5} x + 7 \][/tex]
Substituting [tex]\( y \)[/tex] into the second equation:
[tex]\[ 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \][/tex]
4. Checking the Given Equations:
From the problem, we need to check which of the given equations match this situation.
- The equation [tex]\( 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \)[/tex] matches our derived equation exactly.
Let's simplify it to determine [tex]\( x \)[/tex]:
[tex]\[ 4 \left(\frac{1}{5} x + 7 \right) = 2x - 2 \][/tex]
[tex]\[ \frac{4}{5} x + 28 = 2x - 2 \][/tex]
5. Solving the Equation for [tex]\( x \)[/tex]:
Bring all terms involving [tex]\( x \)[/tex] to one side and constant terms to the other:
[tex]\[ 28 + 2 = 2x - \frac{4}{5} x \][/tex]
[tex]\[ 30 = 2x - \frac{4}{5} x \][/tex]
Combine the [tex]\( x \)[/tex] terms on the right side:
[tex]\[ 30 = \frac{10}{5} x - \frac{4}{5} x \][/tex]
[tex]\[ 30 = \frac{6}{5} x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 30 \cdot\frac{5}{6} \][/tex]
[tex]\[ x = 25 \][/tex]
6. Identifying the Correct Answer:
Peter's age [tex]\( x = 25 \)[/tex] satisfies the equation derived from the problem.
7. Conclusion:
The correct equations representing the situation are:
[tex]\[ y = \frac{1}{5} x + 7 \][/tex]
[tex]\[ 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \][/tex]
The solution to the equations is:
[tex]\[ x = 25 \][/tex]
Thus, the correct equations and the correct value of [tex]\( x \)[/tex] are:
- [tex]\( 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \)[/tex]
- [tex]\( x = 25 \)[/tex]
1. Defining the Variables:
Let [tex]\( x \)[/tex] represent Peter's age in years.
Let [tex]\( y \)[/tex] represent Phil's age in years.
2. Translating the Statements:
- Phil's age is 7 years more than [tex]\(\frac{1}{5}\)[/tex] of Peter's age.
Mathematically, this can be written as:
[tex]\[ y = \frac{1}{5} x + 7 \][/tex]
- Four times Phil's age is 2 years less than twice Peter's age.
Mathematically, this can be written as:
[tex]\[ 4y = 2x - 2 \][/tex]
3. Substituting the First Equation into the Second:
We have [tex]\( y \)[/tex] expressed in terms of [tex]\( x \)[/tex] from the first equation:
[tex]\[ y = \frac{1}{5} x + 7 \][/tex]
Substituting [tex]\( y \)[/tex] into the second equation:
[tex]\[ 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \][/tex]
4. Checking the Given Equations:
From the problem, we need to check which of the given equations match this situation.
- The equation [tex]\( 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \)[/tex] matches our derived equation exactly.
Let's simplify it to determine [tex]\( x \)[/tex]:
[tex]\[ 4 \left(\frac{1}{5} x + 7 \right) = 2x - 2 \][/tex]
[tex]\[ \frac{4}{5} x + 28 = 2x - 2 \][/tex]
5. Solving the Equation for [tex]\( x \)[/tex]:
Bring all terms involving [tex]\( x \)[/tex] to one side and constant terms to the other:
[tex]\[ 28 + 2 = 2x - \frac{4}{5} x \][/tex]
[tex]\[ 30 = 2x - \frac{4}{5} x \][/tex]
Combine the [tex]\( x \)[/tex] terms on the right side:
[tex]\[ 30 = \frac{10}{5} x - \frac{4}{5} x \][/tex]
[tex]\[ 30 = \frac{6}{5} x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 30 \cdot\frac{5}{6} \][/tex]
[tex]\[ x = 25 \][/tex]
6. Identifying the Correct Answer:
Peter's age [tex]\( x = 25 \)[/tex] satisfies the equation derived from the problem.
7. Conclusion:
The correct equations representing the situation are:
[tex]\[ y = \frac{1}{5} x + 7 \][/tex]
[tex]\[ 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \][/tex]
The solution to the equations is:
[tex]\[ x = 25 \][/tex]
Thus, the correct equations and the correct value of [tex]\( x \)[/tex] are:
- [tex]\( 4\left(\frac{1}{5} x + 7\right) = 2x - 2 \)[/tex]
- [tex]\( x = 25 \)[/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
