Perkins Warren
11/16/2023 · Junior High School
\[ \begin{array}{l}0=x^{2}-2 x-3 \\ \text { Write an equation of the line in slope-intercept form that passes through the point }(9,2) \text { and } \\ \text { has a slope of } \frac{1}{3}\end{array} \] \[ \begin{array}{c}y-2=\frac{1}{3}(x-9) \\ y-2=\frac{1}{3} x-3 \\ +2\end{array} \]
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Step-by-step Solution
To find the equation of the line in slope-intercept form that passes through the point \((9, 2)\) and has a slope of \(\frac{1}{3}\), we can start with the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through and \(m\) is the slope. Here, \((x_1, y_1) = (9, 2)\) and \(m = \frac{1}{3}\).
Substituting these values into the point-slope form gives:
\[
y - 2 = \frac{1}{3}(x - 9)
\]
Next, we can simplify this equation to get it into slope-intercept form \(y = mx + b\).
1. Distribute \(\frac{1}{3}\):
\[
y - 2 = \frac{1}{3}x - 3
\]
2. Add \(2\) to both sides to isolate \(y\):
\[
y = \frac{1}{3}x - 3 + 2
\]
3. Simplify the right side:
\[
y = \frac{1}{3}x - 1
\]
Thus, the equation of the line in slope-intercept form is:
\[
\boxed{y = \frac{1}{3}x - 1}
\]
Quick Answer
The equation of the line in slope-intercept form is \(y = \frac{1}{3}x - 1\).
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