Mrs. Teukolsky gave a test that was so difficult that she decided to scale the grades upward. She raised the lowest score , 42 to 60 , and the highest score, 77 to 90 . A linear function that gives a fair way to convert any other test score x to the new score y is

Options

A. `y=-7/6 x +11`
B. `y=-6/7 x +24`
C. `y=6/7x +24`
D. `y=7/6x+11`

Answer

To solve the problem of scaling the test scores, we need to find a linear function that converts the original scores \(x\) to the new scores \(y\). We know two points from the problem:

1. The lowest score: \( (42, 60) \)
2. The highest score: \( (77, 90) \)

### Step 1: Determine the slope of the linear function

The slope \(m\) of a linear function can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (42, 60) \) and \( (77, 90) \):

\[ m = \frac{90 - 60}{77 - 42} = \frac{30}{35} = \frac{6}{7} \]

### Step 2: Use the point-slope form to find the equation

We can use the point-slope form of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

Using point \( (42, 60) \) and the slope \( \frac{6}{7} \):

\[ y - 60 = \frac{6}{7}(x - 42) \]

### Step 3: Simplify the equation

Now, we will simplify the equation:

\[ y - 60 = \frac{6}{7}x - \frac{6}{7} \times 42 \]

Calculating \( \frac{6}{7} \times 42 \):

\[ \frac{6 \times 42}{7} = \frac{252}{7} = 36 \]

So, we have:

\[ y - 60 = \frac{6}{7}x - 36 \]

Now, add 60 to both sides:

\[ y = \frac{6}{7}x + 24 \]

### Step 4: Final equation

Thus, the linear function that converts any test score \(x\) to the new score \(y\) is:

\[ y = \frac{6}{7}x + 24 \]

### Conclusion

The correct option from the given choices is:

**Option 3: \( y = \frac{6}{7}x + 24 \)**.

---