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1. If a line [tex] l_1 [/tex] contains points [tex] P(2, 6) [/tex] and [tex] (0, y) [/tex], find [tex] y [/tex] if [tex] l_1 [/tex] is parallel to [tex] l_2 [/tex] and the slope of [tex] l_2 [/tex] is [tex] \frac{3}{4} [/tex].

Answer :

To solve the problem of finding [tex]\( y \)[/tex] when a line [tex]\( l_1 \)[/tex] is parallel to another line [tex]\( l_2 \)[/tex] with a known slope, follow these steps:

1. Know the given information:
- Point [tex]\( P(2, 6) \)[/tex] is on line [tex]\( l_1 \)[/tex].
- The slope of line [tex]\( l_2 \)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
- Line [tex]\( l_1 \)[/tex] is parallel to line [tex]\( l_2 \)[/tex], so the slope of [tex]\( l_1 \)[/tex] is also [tex]\(\frac{3}{4}\)[/tex].

2. Use the slope formula:
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated as:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]

3. Set up the equation using known values:
- Since the line [tex]\( l_1 \)[/tex] is parallel to [tex]\( l_2 \)[/tex], the slope of [tex]\( l_1 \)[/tex] is also [tex]\(\frac{3}{4}\)[/tex].
- Substitute the known point [tex]\( (2, 6) \)[/tex] for [tex]\((x_1, y_1)\)[/tex] and [tex]\( (0, y) \)[/tex] as [tex]\((x_2, y_2)\)[/tex].
[tex]\[
\frac{y - 6}{0 - 2} = \frac{3}{4}
\][/tex]

4. Solve for [tex]\( y \)[/tex]:
- Simplify the left side of the equation:
[tex]\[
\frac{y - 6}{-2} = \frac{3}{4}
\][/tex]
- Multiply both sides by [tex]\(-2\)[/tex] to isolate [tex]\( y - 6 \)[/tex]:
[tex]\[
y - 6 = \frac{3}{4} \times (-2)
\][/tex]
- Calculate [tex]\(\frac{3}{4} \times (-2)\)[/tex]:
[tex]\[
y - 6 = -\frac{3 \times 2}{4} = -\frac{6}{4} = -\frac{3}{2}
\][/tex]
- Add 6 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 6 - \frac{3}{2}
\][/tex]
- Convert 6 to a fraction with a common denominator of 2:
[tex]\[
y = \frac{12}{2} - \frac{3}{2} = \frac{12 - 3}{2} = \frac{9}{2}
\][/tex]

5. Convert the result to a decimal:
- [tex]\( \frac{9}{2} = 4.5 \)[/tex]

Therefore, the value of [tex]\( y \)[/tex] is [tex]\( 4.5 \)[/tex].

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