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To solve the problem of finding Sharon's and John's ages given the conditions, we can set up a system of equations based on the information provided:
1. Sum of Ages: The sum of Sharon's age ([tex]\(s\)[/tex]) and John's age ([tex]\(j\)[/tex]) is 70.
[tex]\[ s + j = 70 \][/tex]
2. Age Relationship: Sharon is 4 times as old as John.
[tex]\[ s = 4j \][/tex]
We now have the following system of equations:
[tex]\[ \begin{cases} s + j = 70 \\ s = 4j \end{cases} \][/tex]
To find the solution, follow these steps:
1. Substitute the second equation into the first equation:
Substitute [tex]\(s = 4j\)[/tex] into [tex]\(s + j = 70\)[/tex]:
[tex]\[ 4j + j = 70 \][/tex]
2. Combine like terms:
[tex]\[ 5j = 70 \][/tex]
3. Solve for [tex]\(j\)[/tex]:
Divide both sides of the equation by 5:
[tex]\[ j = \frac{70}{5} = 14 \][/tex]
4. Solve for [tex]\(s\)[/tex]:
Substitute [tex]\(j = 14\)[/tex] back into the equation [tex]\(s = 4j\)[/tex]:
[tex]\[ s = 4 \cdot 14 = 56 \][/tex]
Thus, the solution to the system of equations is:
- Sharon's age ([tex]\(s\)[/tex]) is 56 years old.
- John's age ([tex]\(j\)[/tex]) is 14 years old.
### Verification
We can verify our solution by checking that both conditions are met:
1. Sum of Ages:
[tex]\[ s + j = 56 + 14 = 70 \][/tex]
2. Age Relationship:
[tex]\[ s = 56 \quad \text{and} \quad 4j = 4 \cdot 14 = 56 \][/tex]
Both conditions are satisfied, confirming that the solution is correct.
### Graphical Representation
To graph this system of equations, we plot the following lines in the [tex]\(s\)[/tex]-[tex]\(j\)[/tex] plane:
1. The line representing [tex]\(s + j = 70\)[/tex]. This is a straight line with a y-intercept at [tex]\(s = 70\)[/tex] and a slope of [tex]\(-1\)[/tex]:
[tex]\[ s = 70 - j \][/tex]
2. The line representing [tex]\(s = 4j\)[/tex]. This is a straight line passing through the origin with a slope of 4:
[tex]\[ s = 4j \][/tex]
The intersection of these two lines represents the solution to the system of equations, which is the point [tex]\((s, j) = (56, 14)\)[/tex].
In summary, Sharon is 56 years old, and John is 14 years old, satisfying both given conditions.
1. Sum of Ages: The sum of Sharon's age ([tex]\(s\)[/tex]) and John's age ([tex]\(j\)[/tex]) is 70.
[tex]\[ s + j = 70 \][/tex]
2. Age Relationship: Sharon is 4 times as old as John.
[tex]\[ s = 4j \][/tex]
We now have the following system of equations:
[tex]\[ \begin{cases} s + j = 70 \\ s = 4j \end{cases} \][/tex]
To find the solution, follow these steps:
1. Substitute the second equation into the first equation:
Substitute [tex]\(s = 4j\)[/tex] into [tex]\(s + j = 70\)[/tex]:
[tex]\[ 4j + j = 70 \][/tex]
2. Combine like terms:
[tex]\[ 5j = 70 \][/tex]
3. Solve for [tex]\(j\)[/tex]:
Divide both sides of the equation by 5:
[tex]\[ j = \frac{70}{5} = 14 \][/tex]
4. Solve for [tex]\(s\)[/tex]:
Substitute [tex]\(j = 14\)[/tex] back into the equation [tex]\(s = 4j\)[/tex]:
[tex]\[ s = 4 \cdot 14 = 56 \][/tex]
Thus, the solution to the system of equations is:
- Sharon's age ([tex]\(s\)[/tex]) is 56 years old.
- John's age ([tex]\(j\)[/tex]) is 14 years old.
### Verification
We can verify our solution by checking that both conditions are met:
1. Sum of Ages:
[tex]\[ s + j = 56 + 14 = 70 \][/tex]
2. Age Relationship:
[tex]\[ s = 56 \quad \text{and} \quad 4j = 4 \cdot 14 = 56 \][/tex]
Both conditions are satisfied, confirming that the solution is correct.
### Graphical Representation
To graph this system of equations, we plot the following lines in the [tex]\(s\)[/tex]-[tex]\(j\)[/tex] plane:
1. The line representing [tex]\(s + j = 70\)[/tex]. This is a straight line with a y-intercept at [tex]\(s = 70\)[/tex] and a slope of [tex]\(-1\)[/tex]:
[tex]\[ s = 70 - j \][/tex]
2. The line representing [tex]\(s = 4j\)[/tex]. This is a straight line passing through the origin with a slope of 4:
[tex]\[ s = 4j \][/tex]
The intersection of these two lines represents the solution to the system of equations, which is the point [tex]\((s, j) = (56, 14)\)[/tex].
In summary, Sharon is 56 years old, and John is 14 years old, satisfying both given conditions.
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