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]
