To find the equation of the line passing through the point [tex]\((10, 4)\)[/tex] and parallel to the line given by [tex]\(3x + 5y = 8\)[/tex], we'll follow these steps:
### Step 1: Determine the slope of the given line.
The provided line's equation is [tex]\(3x + 5y = 8\)[/tex]. To find the slope, convert this equation into the slope-intercept form, which is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
Starting with the equation:
[tex]\[ 3x + 5y = 8 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ 5y = -3x + 8 \][/tex]
[tex]\[ y = \left(-\frac{3}{5}\right)x + \frac{8}{5} \][/tex]
From this, we see the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -\frac{3}{5} \][/tex]
### Step 2: Utilize the point-slope form of the line equation.
Since we know the slope of the line parallel to [tex]\(3x + 5y = 8\)[/tex] is the same ([tex]\(-\frac{3}{5}\)[/tex]), and the line must pass through the point [tex]\((10, 4)\)[/tex], we use the point-slope form of the equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the point [tex]\((10, 4)\)[/tex] and the slope [tex]\(-\frac{3}{5}\)[/tex]:
[tex]\[ y - 4 = -\frac{3}{5}(x - 10) \][/tex]
### Step 3: Simplify the equation to the slope-intercept form.
Distribute the slope on the right-hand side:
[tex]\[ y - 4 = -\frac{3}{5}x + \frac{30}{5} \][/tex]
[tex]\[ y - 4 = -\frac{3}{5}x + 6 \][/tex]
Add 4 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{3}{5}x + 6 + 4 \][/tex]
[tex]\[ y = -\frac{3}{5}x + 10 \][/tex]
Thus, the equation of the line through point [tex]\((10, 4)\)[/tex] and parallel to the given line [tex]\(3x + 5y = 8\)[/tex] is:
[tex]\[ y = -\frac{3}{5}x + 10 \][/tex]
In terms of entering your answer:
[tex]\[ y = -3/5x + 10 \][/tex]
