To formulate the problem you described in terms of a system of equations, we first set up the equations based on the information given:
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The sum of Sharon's and John's ages is 70: \[ s + j = 70 \]
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John is 4 times as old as Sharon: \[ j = 4s \]
Now, we can replace \(j\) in the first equation with the expression from the second equation:
\[ s + 4s = 70 \]
This simplifies to:
\[ 5s = 70 \]
Dividing both sides by 5 gives:
\[ s = 14 \]
Now substituting \(s = 14\) back into the equation for \(j\):
\[ j = 4s = 4 \times 14 = 56 \]
So Sharon is 14 years old, and John is 56 years old.
Graphical Representation
To graph this system of equations, we can represent it on a coordinate plane where the x-axis represents Sharon's age \(s\) and the y-axis represents John's age \(j\).
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From the first equation \(s + j = 70\), we can rearrange it to solve for \(j\): \[ j = 70 - s \] This line has a y-intercept of 70 and an x-intercept of 70.
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From the second equation \(j = 4s\), we can also plot this line directly. It has a slope of 4 and goes through the origin.
Intersection Point
The solution to the system of equations will be the point where these two lines intersect, which you've already calculated as:
\[ (s, j) = (14, 56) \]
Summary of Graph
- The line for \(s + j = 70\) will slope downward from (0, 70) to (70, 0).
- The line for \(j = 4s\) will slope upward through the origin.
- The intersection point (14, 56) represents the solution to this problem, indicating Sharon is 14 years old and John is 56 years old.
You would see this intersection in the graph you reference, representing the system of equations and the solution visually.
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