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Loren solved the equation [tex]$10=\frac{19}{9}(149)+b[tex]$[/tex] for [tex]$[/tex]b[tex]$[/tex] as part of her work to find the equation of a trend line that passes through the points [tex]$[/tex](1,130)[tex]$[/tex] and [tex]$[/tex](10,149)$[/tex]. What error did Loren make?

A. She should have solved [tex]$10=\frac{9}{19}(149)+b[tex]$[/tex] for [tex]$[/tex]b$[/tex].
B. She should have solved [tex]$1=\frac{19}{9}(130)+b[tex]$[/tex] for [tex]$[/tex]b$[/tex].
C. She should have solved [tex]$149=\frac{19}{9}(10)+b[tex]$[/tex] for [tex]$[/tex]b$[/tex].
D. She should have solved [tex]$130=\frac{9}{19}(1)+b[tex]$[/tex] for [tex]$[/tex]b$[/tex].



Answer :

GinnyAnswer
To find the correct equation of the trend line passing through the points \((1, 130)\) and \((10, 149)\), we need to follow a sequence of logical steps.

1. Calculate the slope \(m\) of the trend line:
- The formula to calculate the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Substituting \((x_1, y_1) = (1, 130)\) and \((x_2, y_2) = (10, 149)\):
[tex]\[ m = \frac{149 - 130}{10 - 1} = \frac{19}{9} \][/tex]

2. Form the equation of the trend line:
- The general form of the equation of the trend line is:
[tex]\[ y = mx + b \][/tex]
- We need to find the value of the y-intercept \(b\). To do this, substitute the slope \(m\) and one of the points into the equation.

3. Choose a point to solve for \(b\):
- Using the point \((10, 149)\) to solve for \(b\), substitute \(x = 10\), \(y = 149\), and \(m = \frac{19}{9}\) into the equation \(y = mx + b\):
[tex]\[ 149 = \left(\frac{19}{9}\right) \cdot 10 + b \][/tex]
- Rearranging to solve for \(b\):
[tex]\[ b = 149 - \left(\frac{19}{9}\right) \cdot 10 \][/tex]

Hence, Loren should solve \(149 = \frac{19}{9}(10) + b\) for \(b\).

Therefore, the correct error that Loren made is:
- She should have solved [tex]\(149 = \frac{19}{9}(10) + b\)[/tex] for [tex]\(b\)[/tex].

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