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Problem:
Find the equation of the line passing through the point (4, -1) that is parallel to the line [tex]\(2x - 3y = 9\)[/tex]. Use a forward slash for fractions (e.g., [tex]\(1/2\)[/tex] for [tex]\(\frac{1}{2}\)[/tex]).

Solution:

Step 1: Find the slope of the line [tex]\(2x - 3y = 9\)[/tex].
[tex]\[ m = \square \][/tex]
[tex]\[ 2/3 \][/tex]

What would the parallel slope be?
[tex]\[ m = \square \][/tex]
[tex]\[ 2/3 \][/tex]

Step 2: Use the slope to find the y-intercept of the parallel line.
[tex]\[ b = \square \][/tex]

Step 3: Write the equation of the line that passes through the point (4, -1) and is parallel to the line [tex]\(2x - 3y = 9\)[/tex].
[tex]\[ y = \square x + \square \][/tex]

Sagot :

GinnyAnswer
Let's work through the problem step by step.

Problem:
Find the equation of the line passing through the point [tex]\((4, -1)\)[/tex] that is parallel to the line [tex]\(2x - 3y = 9\)[/tex].

Solution:

Step 1: Find the slope of the line [tex]\(2x - 3y = 9\)[/tex].

The general form of a line is [tex]\(Ax + By = C\)[/tex]. We can rewrite it as [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.

For the line [tex]\(2x - 3y = 9\)[/tex]:
- [tex]\(A = 2\)[/tex]
- [tex]\(B = -3\)[/tex]

The slope [tex]\(m\)[/tex] is given by [tex]\(m = -A / B\)[/tex].

So,
[tex]\[ m = -\frac{2}{-3} = \frac{2}{3} \][/tex]

Since parallel lines have the same slope, the slope of the line parallel to [tex]\(2x - 3y = 9\)[/tex] is also:

[tex]\[ m = \frac{2}{3} \][/tex]

Step 2: Use the slope to find the [tex]\(y\)[/tex]-intercept of the parallel line.

We need to use the point [tex]\( (4, -1) \)[/tex] and the slope [tex]\( \frac{2}{3} \)[/tex] to find the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]. The equation of the line is in the form [tex]\( y = mx + b \)[/tex].

Substitute the point and the slope into [tex]\( y = mx + b \)[/tex]:

[tex]\[ -1 = \left(\frac{2}{3}\right) \cdot 4 + b \][/tex]

Simplify and solve for [tex]\( b \)[/tex]:

[tex]\[ -1 = \frac{8}{3} + b \][/tex]
[tex]\[ b = -1 - \frac{8}{3} \][/tex]
[tex]\[ b = -\frac{3}{3} - \frac{8}{3} \][/tex]
[tex]\[ b = -\frac{11}{3} \][/tex]

So the [tex]\( y \)[/tex]-intercept is:
[tex]\[ b = -\frac{11}{3} \][/tex]

Step 3: Write the equation of the line that passes through the point [tex]\((4, -1)\)[/tex] and is parallel to the line [tex]\(2x - 3y = 9\)[/tex].

The equation of the parallel line with slope [tex]\( \frac{2}{3} \)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\( -\frac{11}{3} \)[/tex] is in the form:
[tex]\[ y = mx + b \][/tex]

So, substitute [tex]\( m = \frac{2}{3} \)[/tex] and [tex]\( b = -\frac{11}{3} \)[/tex]:

[tex]\[ y = \frac{2}{3}x - \frac{11}{3} \][/tex]

Therefore, the equation of the line is:

[tex]\[ y = \frac{2}{3} x - \frac{11}{3} \][/tex]

So the detailed solution is:

Step 1:
The slope [tex]\( m = \frac{2}{3} \)[/tex]

Step 2:
The [tex]\( y \)[/tex]-intercept [tex]\( b = -\frac{11}{3} \)[/tex]

Step 3:
The equation of the line is:
[tex]\[ y = \frac{2}{3} x - \frac{11}{3} \][/tex]
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