Find a linear equation whose graph is the straight line with the given properties.
through

1,
−2
3

with slope
1
3
y(x) =

To find the linear equation for the straight line that passes through the point \((1, -\frac{2}{3})\) and has a slope of \(\frac{1}{3}\), we can use the point-slope form of the equation of a line. The point-slope form is:

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

where:
- \(m\) is the slope,
- \((x_1, y_1)\) is a point on the line.

Given:
- \(m = \frac{1}{3}\),
- \((x_1, y_1) = (1, -\frac{2}{3})\).

Substitute these values into the point-slope form:

\[
y - \left(-\frac{2}{3}\right) = \frac{1}{3}(x - 1)
\]

Simplifying:

\[
y + \frac{2}{3} = \frac{1}{3}(x - 1)
\]

Now distribute the \(\frac{1}{3}\):

\[
y + \frac{2}{3} = \frac{1}{3}x - \frac{1}{3}
\]

To isolate \(y\), subtract \(\frac{2}{3}\) from both sides:

\[
y = \frac{1}{3}x - \frac{1}{3} - \frac{2}{3}
\]

\[
y = \frac{1}{3}x - 1
\]

Thus, the linear equation is:

\[
y(x) = \frac{1}{3}x - 1
\]

Would you like further details or have any questions?

Here are 5 questions that expand on this topic:
1. How would you find the equation if the slope were different?
2. What is the y-intercept of the line in this equation?
3. How can you graph this equation on the coordinate plane?
4. How do you determine if a point lies on this line?
5. How does the equation change if we use the slope-intercept form directly?

**Tip:** Always check the signs of your slope and point coordinates carefully when using the point-slope form!

Find a linear equation whose graph is the straight line with the given properties: through (1, -2/3) with slope 1/3.

Discover how to find the linear equation for a line with a slope of 1/3 that passes through the point (1, -2/3). This guide explores key concepts such as point-slope and slope-intercept forms, providing a detailed step-by-step solution. Suitable for Grades 8-10.

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