Find the equation of the line passing through the points

left parenthesis negative two thirds comma 3 right parenthesis−23,3

and

left parenthesis negative 4 comma one half right parenthesis−4,12.

Write the equation in​ slope-intercept form.

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Part 1

Select the correct choice below and fill in any answer boxes within your choice.

A.

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

enter your response here.

​(Type your answer in​ slope-intercept form. Use integers or simplified fractions for any numbers in the​ equation.)

B.

The slope is undefined.

To find the equation of a line in **slope-intercept form** \((y = mx + b)\), we need two things:

1. **The slope** \(m\), which is the change in \(y\) divided by the change in \(x\).
2. **The y-intercept** \(b\), which we can find after calculating the slope.

### Step 1: Find the slope \(m\)
Given points \(\left(-\frac{2}{3}, 3\right)\) and \((-4, \frac{1}{2})\), the slope is calculated as:

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

Substituting the given values:

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

Simplifying:

- The numerator:
\[
\frac{1}{2} - 3 = \frac{1}{2} - \frac{6}{2} = -\frac{5}{2}
\]

- The denominator:
\[
-4 + \frac{2}{3} = -\frac{12}{3} + \frac{2}{3} = -\frac{10}{3}
\]

Now, the slope is:
\[
m = \frac{-\frac{5}{2}}{-\frac{10}{3}} = \frac{5}{2} \times \frac{3}{10} = \frac{15}{20} = \frac{3}{4}
\]

### Step 2: Use the slope-intercept form
Now that we have the slope \(m = \frac{3}{4}\), we use one of the points to find the \(y\)-intercept \(b\). Let's use \(\left(-\frac{2}{3}, 3\right)\):

Substitute into the equation \(y = mx + b\):
\[
3 = \frac{3}{4} \times \left(-\frac{2}{3}\right) + b
\]

Simplify:
\[
3 = \frac{3}{4} \times -\frac{2}{3} = -\frac{6}{12} = -\frac{1}{2}
\]

Now solve for \(b\):
\[
3 = -\frac{1}{2} + b \implies b = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}
\]

### Step 3: Write the equation
The equation of the line is:

\[
y = \frac{3}{4}x + \frac{7}{2}
\]

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

\[
y = \frac{3}{4}x + \frac{7}{2}
\]

Would you like further details or have any questions?

### Related questions:
1. How do you find the slope given two points?
2. What does the slope-intercept form of a line represent?
3. How can you calculate the y-intercept once you know the slope?
4. How do you convert an equation into slope-intercept form from standard form?
5. What does it mean if a slope is undefined?

### Tip:
Always simplify fractions where possible when calculating the slope and intercepts.

Find the equation of the line passing through the points (-2/3, 3) and (-4, 1/2). Write the equation in slope-intercept form.

Learn how to find the equation of a line in slope-intercept form passing through the points (-2/3, 3) and (-4, 1/2). This problem covers key algebraic concepts, including slope calculation and the slope-intercept form equation, suitable for students in Grades 8-10.

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