equation of the line through (4/3,-1) and (-3,1/2)

To find the equation of the line passing through the two points \((\frac{4}{3}, -1)\) and \((-3, \frac{1}{2})\), we can use the point-slope formula for the equation of a line:

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

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

### Step 1: Find the Slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substitute the given points \((x_1, y_1) = \left(\frac{4}{3}, -1\right)\) and \((x_2, y_2) = \left(-3, \frac{1}{2}\right)\):

\[
m = \frac{\frac{1}{2} - (-1)}{-3 - \frac{4}{3}}
\]

Simplify the expression step-by-step:

\[
m = \frac{\frac{1}{2} + 1}{-3 - \frac{4}{3}}
\]

\[
m = \frac{\frac{1}{2} + \frac{2}{2}}{-\frac{9}{3} - \frac{4}{3}}
\]

\[
m = \frac{\frac{3}{2}}{-\frac{13}{3}}
\]

To simplify, multiply by the reciprocal:

\[
m = \frac{3}{2} \times \left(-\frac{3}{13}\right) = -\frac{9}{26}
\]

### Step 2: Use the Point-Slope Form

Now that we have the slope \(m = -\frac{9}{26}\), we can use the point-slope form with one of the points, say \(\left(\frac{4}{3}, -1\right)\):

\[
y - (-1) = -\frac{9}{26} \left( x - \frac{4}{3} \right)
\]

Simplify the equation:

\[
y + 1 = -\frac{9}{26} \left( x - \frac{4}{3} \right)
\]

### Step 3: Convert to Slope-Intercept Form

Now, distribute and simplify:

\[
y + 1 = -\frac{9}{26} x + \frac{9}{26} \cdot \frac{4}{3}
\]

\[
y + 1 = -\frac{9}{26} x + \frac{36}{78}
\]

\[
y = -\frac{9}{26} x + \frac{36}{78} - 1
\]

Simplify further:

\[
y = -\frac{9}{26} x - \frac{42}{78}
\]

Or,

\[
y = -\frac{9}{26} x - \frac{14}{26}
\]

### Final Equation

The equation of the line in slope-intercept form is:

\[
y = -\frac{9}{26} x - \frac{14}{26}
\]

or,

\[
y = -\frac{9}{26} x - \frac{7}{13}.
\]

Would you like to know more details or have any questions about this? 

Here are 5 related questions you might find interesting:
1. How do you find the equation of a line given two points in different forms (standard or point-slope)?
2. What is the significance of the slope in determining the behavior of a line?
3. How can we find the perpendicular or parallel line equations to a given line?
4. How can you convert from slope-intercept form to standard form?
5. What are some real-life applications of finding the equation of a line?

**Tip:** Always double-check the arithmetic when calculating the slope to avoid simple mistakes that can lead to incorrect line equations!

Learn how to find the equation of a line passing through the points (4/3, -1) and (-3, 1/2) using the point-slope formula and slope-intercept form. This detailed solution covers the calculation of slope, application of formulas, and simplification steps involved. Suitable for high school students.

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